ZanderMy quickest deal (not the most profitable) took me 2 visits to the owner (1 block from my home), one network meeting and 3 days from purchase to sale. 

Should I give the bank a call or pay it a visit?

I would visit the register of deeds or call a title company and find out if there is any merit to the claims being made. 

If your staying in the house long term, and the condition of the current furnace isn't great, plan ahead for the repair and do it on your terms, not when it becomes an emergency.

Do I go ahead with deal or cancel it?

It's also wise of both of you to sit with an attorney and work through the scenarios ahead of time, despite knowing each other so well.

I will go ahead and adjust it.

I don't care who they have over there as long as they pay the rent on time and take care of the property.Imagine your bank harassing you because you allowed a family member or friend to stay with you for a period of time....or even move in with you.Pick and choose your battles, but if you want to die on that hill then go ahead. 

Sometimes they will lease housing for 3-6 months for visiting doctors/nurses/professors and are willing to pay a higher rent.Airbnb/short term rental if local regulations are allowed and doable.

It amazes me that tenants often want to rent the nicest place they can qualify for instead of trying to cut their largest expense so they can get ahead.