I want to tear down my house and rebuild it. I heard if you leave an original structure in tact like a wall or fireplace, it is considered a remodel. Say if I bought the house 2 years ago for $400K and rebuild costs $200K (essentially a new house)... how does the city assess the new value? Also, if i have a mortgage, does the bank have to approve the rebuild?

This depends on your local code enforcement office and the local codes.

Yes, the bank will be unhappy if you tear down their collateral.