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).
Can you explain your rationality for this? You're still allowing the bank to have senior ownership of your house except... for less money.
If you default, they keep your house, no matter what the debt-to-value is.
Like, I understand saying "I couldn't make my mortgage payments so the bank took the house, but I guess they did pay for 90% of it".
But this seems like all the insecurity of a mortgage for a lot less money?