I'm going to take a crack at this one, as 'non-earthlike-energy for native life forms on exoplanets' is something that has interested me ever since I found out about the life here on earth, around volcanic vents under the sea, that use the earth's own heat for energy instead of the photosynthesis cycle (something widely considered to be impossible before its discovery). I may add more after I read other's answers for inspiration, but here's what I have off the top of my head (Disclaimer that I have no scientific experience to back any of this on, beyond entry level college biology class, and my own layman's 'research' prompted by my curiosity):
Stress: I would expect this to be the most primitive form of life in this scenario, analogous to single-celled organisms here on Earth, and possibly the simplest of multicellular life. A fast spinning moon could allow single celled kinetoautotrophs to use the gravitational changes felt from the gravity of the planet, both in intensity and direction, as the source of the stress. The primary environmental factor they would need for this 'niche' would be a solid surface to live on, to provide a resistance force against the gravitational one. Evolution, from there in to multicellular organisms, could yield a structure similar to a sea sponge or moss clumps, spherical or semi-spherical except where in contact with irregularities in the surrounding surfaces, and with cells at the bottom adapted to take advantage of higher stressses (from the increased weight above them) at lower ranges of motion, while cells above adapted to lower stresses and higher ranges of motion. The next evolution would be more advanced structures on the bottom of organisms to grip the surface to prevent tipping over during growth, and thus avoiding ending up with the wrong specialized cells in the wrong orientations, and to extract nutrients more efficiently from the hard surface or any incidental liquid that might be present. Minor variations would exist for different climates. Wider and flatter in high wind areas or where flooding is a concern, taller and more conical in calmer areas. And that's where I think evolution would branch to adapt to more varied surfaces, and based on environmental conditions related to the other sources of energy, wind, rain, tides (sound adaptations come later).
Wind: To me, this seems like the first natural adaptation to an alternate form of kinetic energy, simply due to the fact that it would be so wide-spread across the surface of the moon, regardless of the presence or absence of standing, flowing, or falling liquid, and especially if my assumption (mentioned in my section on stress energy) of a fast rotation speed is accepted since this rotation would also cause (relatively, compared to slower rotation speeds) more or less constant winds with and high average speeds for winds. Structural adaptations for this would vary widely, depending on surface, and also depending on prevailing wind conditions like speed (fast or slow) and consistency (steady or gusty). For broken surfaces (gravel, sand, soil, etc) root-like structures would be likely, to help keep from being blown over, and to provide a stable foundation to allow the possibility of more and greater vertical growth. For more unbroken solid surfaces, I would expect a horizontal variation of what is seen on wall-climbing vine plants, a network of 'branches' sent out from the base to seek out any of the relatively rare nooks and crannies that could be used for anchorage, and also to simply provide a wide base for support in the absence of significant or sufficient anchor points in the surface itself. Now, on to the energy collecting structures. Simpler forms of life would have structures that appear, superficially, like blades of grass, or a very short stem (stems with wide separation if multiple are present in a single organism) with a single leaf-like or fan-like (like an accordion-folding fan) structure at the end of each stem. The grass-like blades (more common in steady winds) could be air-foil shaped (think an airplane wing tipped up verticle) causing a bending motion (and subsequent stress for energy) in response to the wind, or have a cross section shaped like the letter "C" so that as wind fills the gap it will spill out of one side, causing a twisting motion (and subsequent stress for energy) before rebounding and catching more air which spills out the opposite side, oscillating it back and forth repeatedly (like a ribbon pulled semi-tight in stiff breeze). Fan structures (more common in areas with more calm air and intermittent stronger gusts or windstorms) could vary more widely, portions of a circle (nearly full circle, semi circle, quarter circle) for maximum surface area with minimal mass to take advantage of every slight change in the air movement in regions of low prevailing wind speeds and low intensity/frequency gusts and low intensity/frequency windstorms, to more exotic variations like very thin and fibrous webs (like dandelion fluff, but in any and all shapes) or a 'kite' with a horizontally oriented airfoil shape at the top of the stem to intentionally 'lift' and stretch the stem. More advanced varieties would evolve other ways to increase their surface area impacted by the wind, and increase strength (probably through thickness) of stem (trunk) and anchorage to withstand the compounded forces received by the additional impacted surface area. To do this, many would literally branch out like earth trees except that on earth optimum branching direction is perpendicular to the direction of sunlight (which averages to vertical from the surface of the earth, causing branching horizontally, causing plants to generally have a round shape when viewed from the top down) while optimal branching direction for wind gathering plants on this moon would be perpendicular to the prevailing wind direction, so they would generally branch north, south, and vertical, since prevailing wind direction would be either east to west, or west to east, depending on the direction of the rotation of the moon. So viewed from the top down, the branching plants would primarily look like long thin lines, slightly tapered at either end. Viewed from the north or south, they would again look long and thin, and tapered from top to bottom. Viewed from the east or west, they would most likely look like a rounded fan or semi-circle, or the very advanced ones might evolve strong enough anchors and trunks to make more significant vertical gains and look like a circle on the end of the stem/trunk. The advanced species that don't branch might evolve much wider bases perpendicular to the wind, and then fan out vertically, like giant versions of leaves. Others might send out long thin streaming threads from elevated branches to whip in the wind at the ends for maximum movement and energy gains. To compete with other organisms in the direct vicinity, any of these variations would likely use their base/root/anchor structures to seek out competition toward the prevailing wind direction, since any organism in 'front' of them (relative to wind direction) could block their source of energy, while organisms behind them would only be able to compete if they can also send something forward to 'attack' organisms. Once a competitor is identified, the organism behind would seek to either cut off the base or cover the face of the competitor. The most desirable reproduction strategies would be anything that sends offspring in the direction of the prevailing wind, so seeds in the air would be counterproductive, unless there is a band of suitable environment that is a complete circle around the moon (not likely). So budding up from anchors out in front of (though likely not directly in front of) the organism is a likely, creating colonies of organisms with members of a generation generally spread out north to south from each other, and members of newer generations in front of them, though staggered/offset north to south of the older generation behind them.
Rain: This one is the one I think is the most interesting. A quick Google search got me estimates of up to about 14% of Earth's land area was once (pre-deforestation) covered by rain forest. Adjusted however you like for the total land area (as opposed to ocean) of this moon, and it's still a relatively rare thing, and the highest frequency I could find for rainy days per year was 243, in Belem, Brazil, in the Amazon rain forest. That's almost exactly 2 days out of 3, in the place where it rains the most frequently that I could find (not the highest amount of water volume, but the most reliable rain). This makes rain a very unreliable energy source (in general, on a the scale of the entire surface area of a moon[planetary? scale {but it's not a planet} moon-itary scale?]), except in the most ideal climates. So this type of life, with adaptations for this energy source, would likely be both relatively rare and relatively isolated. I see simple life forms evolving from the sponge-form or mossclump-form bu sending out horizontal (Like the branching of trees and plants on Earth spreading horizontally to catch the vertical [by average] sunlight) appendages from a ring near the half way point(ring) between their base and their peak to catch the energy from the vertical (again, average) falling rain. Somewhere near halfway between top and bottom is because too low and there is no room for downward flex after impact from the raindrops, and too high and it would be pointing straight up and there would not be room for enough appendages making it hard to have enough surface area to catch enough rain to be useful. In intermediately advanced life, the spongy/moss base evolves in to a more specialized root/anchor structure, which spreads out more efficiently, either by flattening like a disc, or by separating in to a more branching network, and the appendages become more specialized as well, some becoming dedicated support structures (stems/trunks), while others specialize at catching and using the rain energy by either increasing in number while thinning out (think very long thin stiff grass growing sideways instead of vertical), or by forming wider leaf-like shapes. The more advanced would combine the two, having very small fuzzy hairlike structure on their leaves that would react to the flowing of the rain off of the leave after impact, while the larger leaf structure focuses on harnessing the energy of the impact itself. The most advanced appendages would be even more highly specialized, forming funnel shapes (complete with openings at the bottom) with their largest leaves, and lining the inside surfaces with fine hairs, creating artificial currents at the bottoms of the funnels so that collected rain could provide longer term energy as it flows slowly through the end of the funnel even after the rain has stopped. Competition with local organisms would lead to both horizontal (cover up the neighboring organism) and vertical (get high enough that you can cover the neighboring organism) adaptations. These would likely be the only 'true' 'trees' on the planet, though the only thing that might encourage any significant height would be the low gravity on the moon. If the gravity is not enough of a factor, then even these trees would be relatively short compared to earth's trees. Being covered up by a competing organism could lead to special adaptations to catching residual rainfall coming off the leaves of the organisms above, such as extremely large individual leaves funneling in to relatively complex and efficient funneling systems. Unfortunately, nothing specific comes to mind for reproduction specializations for this energy form.
Tide: I see two main methods here, sheer ocean surface level changes, and water flow changes like currents and waves. The adaptations for currents and waves are more likely to be more primitive than adaptations for ocean levels, because the immediate brute force of the waves are likely to have a more direct and immediately impactful influence on the sponge/moss close to the shoreline than a relatively calm and slow rise of the tide. The first adaptations would be the strongest anchoring system so far, to avoid being dislodged entirely and lost to the depths, and appendages to move with the flow of the waves and harvest energy from them. I see two main types of appendages forming here: the first is very stiff and strong, to take a beating from waves without breaking, while flexing just enough to create great internal stresses for energy collection. probably starting our as relatively straight spines like a sea urchin's, and later would be more complex lattices like fan coral but more flexible. The other would be VERY flexible, like fine strands of moss, or flowing kelp leaves, making use of movement by gathering less energy per movement, but making up for it by moving more often and in more directions. The most advanced species in this group take advantage of the tides directly. These are the ones that competed with the other wave dwellers originally, but got their start in the deeper side of the coastal waters and managed to survive despite being distanced from the most energetic areas of the waves nearer the shoreline. To avoid sinking too deep below the level where the wave movement wasn't sufficient, they developed gas filled bladders to keep their energy capturing appendages up nearer the surface where the wave energy is stronger. This increased pressure on their anchoring structures, leading to improved anchoring structures. This allowed access to increased depth while maintaining access to energy. This created a cycle of evolution, increased depth > increased bladder > increased anchoring > repeat. Eventually the depth achieved was so great, that sufficient wave energy was out of reach to support the nergy requirements of the stalk structure between the anchor and the energy capturing leaves. The next adaptation at this point was for the stalk structure itself to become an energy producer, from the stretch stress between anchor and gas bladder. Further adaptation moved this stress from secondary to primary, and the leaf structures become vestigial, or disappear entirely. At the same time, the anchor, stalk, and gas bladder all become exagerated until the most advanced species is an enormous bladder that floats on the surface of the ocean at low tide, and is mostly or completely submerged at high tide. This is connected to a very strong, thick, stalk which receives massive stress energy from the bladder pulling up and the anchor structure holding it down. Competition would be mostly based on surface area under the water, for anchor points, individual organisms would seek to cover as much of the available nooks and crannies within their reach, to ensure the best hold. I see no reason that reproduction in this oceanic environment would follow a pattern any different than earth's oceans, so budding/self-cloning, and releasing egg/sperm directly in to the water on regular cycles, are likely options regardless of the specific oceanic region or adaptation of the rest of the organisms body type.
Sound: This would be the most advanced individual adaptation of any of the groups, and would apply to all of them, from the sponge/moss to the wind, rain, and tide, types. In other words, the adaptation wouldn't be specific to any of the groups, but any species within the group that has this adaptation would be the among the most advanced species in their respective groups. This mechanism actually exists in the animal kingdom on earth, and the human inner-ear is a good example. It has hair-like structures (an adaptation I already mentioned for other types) that convert movement from sound waves to electrical signals. The sponge and moss clump species could directly produce these hairlike structures externally, and the sponges could also have them inside the cavities that already existed in their more primitive relatives. Wind-specialists could incorporate them on their surfaces as well, and the most advanced would be the ones with ancestors that had already specialized to have a higher number of leaves/appendages with smaller surface structures, thus allowing their more advanced descendants the benefit of having more surface points from which to generate those hairlike structures. Rain-specialists would probably evolve this adaptation before the others, as the movement of liquid inside confined spaces is already part of their initial strategy, so this would just close the loop for them. Tidal-specialists could sprout those hairs from any part of their structures, since they are (almost) entirely submerged (almost) all their lives. The massive bladders of the deepest tidal species could have those hairs both inside and outside the bladders, to take advantage of sound both through air above (at low tide) and below.
my own group: A hybrid type, not mentioned by the OP, seems likely to me. If it rains, and there are oceans, then there are almost certainly rivers that take the rain back to the ocean. Specialist species, hybrids(in form if not literally) between rain/wind/tidal, could anchor to the banks of rivers(easier to anchor on land, where water isn't trying to rush the seedling away from any anchor points), and reach appendages in to the current where they act like a combination of wave/wind energy collector appendages. Advanced variations could still use funnel forms like the rain collectors, but submerged. The Sound collecting hairs would not function on 'sound', specifically, while submerged as the current of the water would likely be universally more efficient than sound collection in this environment, but the surface structures could use them.