The bank doesn't "keep" your house, they foreclose on it and sell it. Once they sell your home, any liens and associated costs are satisfied and the remainder of the money is given to the borrower.
No, any value outside of what is owed to the bank.
If you take out a 300k mortgage and pay it down to 150k principle then default, the bank sells it for 250k and gives you 100k (minus associated costs).