(1) It is lawful to give a house on rent to the bank but it is better to avoid.
(2) The Islamic ruling regarding the proposal is that at the time of fixing it the religiousness, nobility and trustfulness should be given the first priority. A boy who earns pure and halal earnings though he is weak financially is better than one who earns impure and unlawful earnings. Likewise, such a girl is better who is religious minded, modest and obedient to her husband. The income of bank is not fully pure so if you could find a proposal better than that person working in the bank, then you should opt for the proposal.
(3) The ruling regarding eating and drinking at the house of any Muslim and accepting his invitation is that if the feast is done with interest money then it is not lawful to participate in the feast at all. And if it is known that the feast is with halal money then it is allowed to join the feast. But, if it is not known definitely then it will be seen whether the major part of his income is halal or haram. If his major income is haram attending the feast is not lawful and if major part is halal participating in the feast will be allowable. And if it is not known whether the most income is halal or haram, it will not be right to enquire about his income being halal or haram; because it will imply to hurt him and distrust him. The holy Prophet Muhammad (صلى الله عليه وسلم) forbade from it. (ترمذى باب ما جاء فى تعظيم المؤمن، حديث نمبر 2023)
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband