I read in one of answers about foreclosure, that one of the things the mortgage company does as a process of foreclosure is inspect the house. If this is done when is it done before or after the " owner" moves out? What is the procedure and what type of notice is the "owner"
Given, if done before an order to vacate?
What are the "owner's" rights?

It all depends when the lower moves out. It will be done prior to the actual foreclosure procedures, often at the start, the mortgage company hires a real estate company to do the inspection. You don't have to let them in of course, although your mortgage agreement most likely says you have to, but you are already in breach by not paying, so another breach is really not important. If you don't it will just most likely speed up the foreclosure process and make the mortgage company less likely to try and work with you.