Unfortunately I can't post a comment since I'm new here and lack the reputation, so I hope posting an answer would be ok despite not fully solving the puzzle. I'll try to give minimal hints to help you progress bit by bit, will keep giving more if you get stuck again.
For now: Based on hint #9, Adam will live at a lower floor than Mr Gates. You've deduced that Adam either lives in the 2nd or 4th floor, so there is no way Mr Gates can live in the 1st or 2nd and still fulfill this rule. Addingt his information to your current grid, you can deduce the surname of the person living in the first floor.
Edit: Sadly rule #5 also states that the one surnamed Banes cannot live in the first floor, as Barbara must live lower than them. You've made a mistake in your grid somewhere, I'm afraid.