So you are somewhere there. I will rewrite each of your statements:
FE: controls for time-invariant differences between groups. It likely has less OVB than RE because it removes any of the omitted variation due to time-constant factors. Then the remaining OVB in an FE estimation method can only come from time-varying factors.
FE: The FE estimation method has 2 identical methods: Demean each variable from its group average across time. Or, add a dummy variable for each group.
RE: As RE does not remove the time-invariant variation, you can estimate time-constant variables in an RE model. FE eliminates that variation from estimation.
assumption of RE: the time-invariant heterogeneity between groups is uncorrelated with the error term. FE does not make this assumption and thus we remove that variation. BOTH of these models assume that the error term is uncorrelated with the observable predictors to be consistently estimatable (not sure if that's a word).
what are the limitations to FE and what to RE models? - FE controls for a lot of potential OVB, but by doing so it limits what you can estimate. For example you cannot estimate the effect of gender on something in an FE model. RE models are more relaxed in that you can do that, and they are more efficient (smaller SEs) but they risk more OVB.
when would I prefer RE over FE, when the other way round? - Generally, FE is a safer method and you should only prefer RE if you are confident that the assumptions hold. Some people think a Hausman test can help in determining which you should use.
Last note: Modeling with an FE estimation method does NOT eliminate all OVB. Any time-varying factors you do not adequately control for can affect your results. So while FE is safer than RE, if you care about the consistency of your coefficients, be careful with any estimation method that is not quasi-experimental like 2SLS.