Do Boltzmann brain thought experiments suggest literally anything can form randomly?
What are the limitations to what random fluctuations can form? Literally any physical, material object?
Lastly, I am curious as to how this compares to an object with a high degree of complexity always existing. For example, is a bike more likely to always exist for no reason than it being created by random fluctuations? Or is it less likely?
A very handy way of looking at the statistical mechanism of a system is by looking at its so-called phase space (read as state space).
Consider a microstate $X = (q_1, . . . , q_N, p_1, . . . , p_N)$ of a classical system that consists of a $N$ particles. Here, $q, p$ represents the position and momentum degree of freedom of the particles. Naturally, the dimension of this phase space is $2N$.
It is assumed that every microstate (a point in this phase space) is equally likely to be attained by a physical system. Each such specification of $X$ determines a macrostate. Now, if we wait for a long period of time, then every microstate would occur at least once. Such systems are called ergodic.
So yes, these $N$ particles can assume any microstate, which includes a brain popping out of nowhere (although with a vanishing probability). However, there are more microstates corresponding to certain macrostates than others. Since each microstate is equally likely, therefore the physical system of $N$ particles is more likely to be found in certain macrostates than others. These more likely macrostates are the states which maximise entropy solely because they contain more microstates and hence occupy a larger volume of the phase space.
But...
Above, I assumed that all the $N$ particles were free to move around (i.e., there were no constraints on their dynamics). Imposing constraints (depending on the details of the dynamics, e.g., due to internal forces or conservation of total energy which then sets an upper limit on the $p$'s) reduces the volume of the phase space and, therefore, makes certain regions of the original phase space inaccessible. As a result, only those states could be reached that satisfy the imposed constraints. As such, not every thing can form randomly.
For more, I redirect you to this nice article.
Anything that doesn't violate the laws of physics/logic can form. And even for things that violate the laws of physics and/or logic, the memory of having perceived them may still be able to form. So pretty much any phenomenon can form (that is, any phenomenological experience). For instance, the laws of physics say that a region of space with mass but no spacetime curvature is impossible, but it's still possible to have experiences such that it appears that there is mass but not curvature.
As to your question about a bike, the number of microstates in which a bike exists for a short time is larger than the number of microstates in which a bike exists for a very long time, so the former is more likely. On a short enough time span, though, it would be more likely to exist for a longer period of time. For instance, the probability of a bike existing for a microsecond would be larger than the probability of it existing for only a nanosecond. Most of the probability space "near" a state of bike existing also consists of a bike existing. An analogy would be drawing a circle on the ground and putting a feather inside the circle. On the scale of microseconds, the most likely state is that the feather is still inside the circle. On the scale of a year, it's very unlikely that the feather is still inside.
