If the home is truly an REO then it is owned by the bank and is no longer owned by the former owner . The former owner would have no right to rent the property and would be committing fraud if they tried to do so. If your question is whether an owner can rent out their house knowing it is in foreclosure then the answer is yes, they are legally within their rights to do so. Whether it is ethical is another question all together.
Ummmm . . . no. If it's an REO, then the house is owned by the bank. It's not owned by an individual, a "home owner," presumably the person who owned it before the bank took it over. That'd be fraud. Could a person sell a car that he didn't own? Could a person give away furniture owned by a neighbor?
So, no. An individual can't rent out a house (unless acting as an agent, which is not the case here) owned by someone else.
If the home has actually been foreclosed on, no. The house belongs to the bank. To do so would be defrauding the tenant and you could get in a lot of trouble. If the home is simply in the pre-foreclosure state, then the answer is yes. You still own the home and you can do what you want, but you still have to be careful not to do anything that would be considered fraud against the tenant.