I'm sorry for the delay in my reply. Apparently, it never went through. (Attempt #3)
1) The membrane always fluctuates (due to thermal fluctuations, electrostatic interactions between charge lipid head groups, etc.), regardless of random or homogeneous distributions. The curvature is likely to change, and values of curvature will fluctuate even more if the system is not equilibrated, to the extent that the results obtained may be unrealistic. Hence the importance of reaching an equilibrium state and averaging values.
Ideally, we should run long MD simulations (to reach equilibrium) and run several replicas (to have decent sampling) to obtain a meaningful average (over time and over replicas). But we should be realistic in our research too.
2) If the distribution is homogeneous, your system is equilibrated, and you are calculating an average, that's expected. This is also related to the first Q/A.
3-4) I probably can't answer that. I would assume you are performing a curvature analysis where negative or positive mean/Gaussian curvature has a meaning for you.
Best,
Estefania