Is it permissible to build a house on top of a house built with money borrowed from an interest-based bank, whether that is with the father's free consent, or by purchasing the roof from him with lawful money?

There is no harm in building on the roof of your father's house, whether he gifted you the roof or you paid him compensation for it. It is not forbidden for the father to benefit from a house built with an interest-based loan, because the prohibition of the loan pertains to his own responsibility, not to the specific item for which it was used. He must hasten to repent.