When a land lord is undergoing a loan mod. on a rental and not paying the bank can he collect rent?

Yes, of course you can. The property belongs to you and as long as someone is living in your property they owe you rent. What you do with your income is of no relevance at all.

Yes you must keep paying him. on a side note a lot of mortgage have a clause that would require the landlord to pass the rent onto the bank. Still has nothing to do with you so keep paying or he can kick you out!

Yes. Your obligation to pay rent is not contingent on him fulfilling his obligations with the bank. As long as he owns the house, your contract and obligations are 100% valid and enforceable.

Yes. Your contract with your landlord has absolutely nothing to do with his contract with his lender.

good luck!