Can a person rent out their house if it is being forclosed on and can they have a bonding legal rental agreement?

The foreclosure process does not nullify an otherwise binding rental agreement. I said "process" because the homeowner can stop a foreclosure at anytime by curing the default (typically by paying the past due amount owed). The actual foreclosure in Texas would cancel any inferior rental agreements. If you are a tenant of a property that is in foreclosure, there is a new law that provides additional time to vacate the property in the event the foreclosure actually happens. You can read the new law here:

http://www.nlihc.org/doc/701-704-Public-Law-111-22.pdf

This new law does not affect you ability to sue the landlord in the event you are not allowed to stay in the property for the full term of the rental agreement.