Your intuition is correct for the first question: the posterior is the normalized product of the likelihood and prior, so you should not expect the posterior to necessarily be the same form as the prior (except in some conjugate cases).
The skew in the posterior of some of the families may be fine, particularly if your dataset is small. It suggests some residual uncertainty about the location of those parameter values. You did not state how many tuning and post-tuning samples you drew, but make sure your MC error is an order of magnitude or more smaller than the posterior standard deviation.