The English translation should read "Have you paid this month's /house/ rent?"
I put, "Have you paid the rent this month?" they didnt like that formulation either :-P
Yes, it's unnatural to specifiy that it's on the house, because what else would rent be on? And it should be on the house or apartment, rather than "of".
Yep, that's exactly what I put. We don't usually use these long constructions in the form "the X of the X"