This is a continuation of the following question. The previous link was related to rejection sampling. This is related to MCMC.
1a. As far as I understand, the rejection sampling will not overestimate the density as long as $c\cdot q(x) > p(x)$. This is inherent in the acceptance ratio. But as far as I understand, the Metropolis–Hastings algorithm may underestimate or overestimate the density as $q(x|x')$ is not always $> p(x)$. Am I correct? If so how to avoid this?
1b. If $q(x|x')$ is not always $> p(x)$, how exactly does M-H also estimate the target density correctly?
How do I choose a good proposal in case of Metropolis–Hastings? I usually use Gaussian with std=1 and mean centered at the previous sample. It works well in most cases. I just play around with the std. Is this a good proposal?
Rejection sampling gives exact density where as MH gives approximate density. Am I correct?
Is independent Metropolis–Hastings the same as rejection sampling, since the proposal is independent of the previous sample? If so, do we always have to take care to choose proposal > target in case of independent MH?
This is a general sampling question. In most of the materials that I have read, I come across the statement that sampling is usually necessary to sample from posterior density? Why do they stress the word posterior. Why not prior or likelihood. Is there a specific reason. Can you give some examples?