I ran a simple normal regression in rstan with some informative priors. My data has heteroskedasticity and would like to fix the same. However, am new to bayesian regression and rstan. My questions are 1) Can I use the mean of the estimates of coefficients for forecasting out of sample? 2) How do I find the standard error of my regression? Can I still do sigma^2*(X'X)^-1? 3) can I just plot the result of the residuals of regression against fitted values to see the test for heteroskedasticity?
Your first question is answered in the Prediction based on bayesian model thread. Notice that you can make two different kinds of predictions: you can predict the distribution of outcome and the point estimate. If you need a point estimate than you can take mean, median, mode (i.e. maximum a posteriori estimate) etc., depending on your needs.
As about standard errors, what kind of errors do you need and what for? With Bayesian model you will obtain posterior distribution of your parameters and posterior predictive distribution of your outcome. With those distributions you can easily obtain interval estimates (see credible intervals). In fact Bayesian credible intervals give you much more information about the distribution of your outcomes then confidence intervals as discussed in here.