The moon would be similar to Earth in size and composition and the gas giant comparable to neptune or uranus.
Will the periods of eclipse of the the giant have an effect on climate?
Very little compared to everything else.
Your moon will almost certainly be tidally locked, meaning that one hemisphere will always face its parent gas giant, and one half will always face away. It probably won't have a moon as big as Earth's (due to gravitational interference from the gas giant) and that means that the huge angular momentum the Moon brings to our system will be absent and so tidal locking will inevitably occur.
Similarly, the geodynamo that drives Earth's magnetic field may well have cooled to the point where your moon's magnetic field is reduced or absent compared to Earth, because without Earth's spin or tidal forces induced by the Moon, cooling will occur much more quickly than it has on Earth itself. Part of the reason that Mars lost its atmosphere was the lack of a magnetic field, but we know that Titan has kept its atmosphere at least in part due to protection from Saturn's magnetosphere.
Lets say your gas giant is like Uranus. Uranus' magnetopause is at a distance of ~460000 km from its centre. If we put your world at ~400000km, it could have a 7.6 Earth-day long orbital period, which could be called either its day or its month, up to you. That means that its periods of daylight will be long and hot, and periods of night-time will be long and cold. A thick atmosphere with superrotation (as found on Venus and Titan) will help keep heat in and distribute it evenly across the world's surface. It could be quite windy!
Preventing a runaway greenhouse effect might be difficult in these circumstances. I'm not sure how it could be done. You might end up with a hot, wet, pressure-cooker like water world.
The gas giant will have an angular diameter of ~14°41'... that's pretty big, nearly 30 times wider than the full moon seen from Earth. It will have an apparent magnitude of -19, more than 300 times brighter than the full moon. The gas-giant facing hemisphere of your world will have bright nights every night (but for cloud cover), and wildlife will adapt accordingly.
What about the tidal force?
This is usually a hideously difficult question to answer well, but as I'm assuming that your world will already be tidally locked it does simplify some things.
Real-life moons like Io interact with other moons in their local system (like Ganymede and Europa) which tweaks their orbit to be non-circular, resulting in tidal heating. Your moon is absolutely vast by comparison, making it a) hard to tweak and b) even less likely to be near any other moons. Without these peturbations, tidal effects will circularise you moon's orbit in a relatively short period of time, astronomically speaking.
It can therefore be reasonably said that there are no tidal effects, and indeed your moon would in fact experience fewer tidal effects than the Earth does from its own Moon (because the Earth rotates, and your world does not). This means the seas will have solar tides only, for example.
Will being closer to the sun or farther from it on the other side of the giant have any repercussions?
Not nearly as much as you might think! If your gas giant was the size of Uranus, and it was as far from a star like the Sun as Earth is now, your moon would probably have a maximum stable orbit radius of no more than 1.8 million kilometres. If its orbital plane lay in the same plane as the gas giant's orbit about the star, you'll get a difference of solar irradiance of ~5% between closest and furthest points. Earth itself has its perihelion some 5 million km closer to the sun than its aphelion, and it makes less difference than the axial tilt to our experience of winter and summer.
Footnote: here's my thinking on tidal locking.
The time for a body to become tidally locked can be approximated by $$T_{lock} \approx {\omega a^6 I Q \over 3Gm_p^2 k_2 r^5}$$ where $\omega$ is Earth's spin rate (~2π rad/day), $a$ the orbital semimajor axis, $I$ is the moment of inertia (which is 0.331 x the mass of the Earth x the radius of the Earth squared), $Q$ is the dissipation function, $G$ is the gravitational constant, $m_p$ is the mass of Uranus, $k_2$ is the Love number of the Earth and $r$ is Earth's radius. For Earth, Q appears to be about 100 and $k_2$ appears to be about 0.308.
Your moon will not be orbiting beyond more than about half of the gas giant's Hill Radius, which for Uranus at 1AU from the Sun gives a limit of ~1.8 million km and for Neptune ~1.9 million km.
This gives a tidal locking timescale of ~130 million years for either case... sufficiently quickly that your moon could be doomed to a Mars like fate as its geodynamo cools, the magnetic field weakens and the solar wind blows the atmosphere off. Maybe the atmosphere will remain (as it has on Venus, though you might not want to live there) but I don't think that the tidal locking can be avoided in your scenario.
A closer moon will lock much faster, but will be protected from the solar wind. It'll also get much brighter nights on the planet-facing side, with a generally awesome view. What's not to like?
I suggest that yu use the Search on Worldbuilding feature at the top of the worldbulding bar and search for something like "Habitable Moons".
I also suggest that you search for articles about the possibility of Habitable exomoons by Rene Heller & others.
https://faculty.washington.edu/rkb9/publications/hb13.pdf[1]
