To get the prediction variance in a Bayesian treatment, you integrate over the posterior of the parameters - surely computing or approximating the posterior counts as considering parameter variance?
An estimator (of a parameter) is a random variable, as it is a function of random variables, however this depends only on the data distribution, there is no other implicit distribution on which it depends.
For instance, the distribution for the maximum likelihood estimator of the mean of a normal distribution is normally distributed, however this does not imply that the mean parameter has a normal prior, it has no prior, as it is a fixed quantity.
Yes
> you also make assumptions about the parameter distribution
No
> your variance estimate only holds asymptotically
Don't follow