(You may trust the person, but verify all the info for yourself because they may not fully understand what they're telling you).  

That being said, I'm usually pretty fair and call landlord references, check pay stubs, and review application to see if they meet criteria, and before I'm about to accept, I request the credit/ criminal check to verify all of the information is correct.

Or worse yet what if the person is "willing to pay above market rent" because they have bad credit (doesn't intend to pay), no verifiable income (can't pay), or a track record for trashing rentals?

@Atwan Kwan Did you physically inspect the property and verify that there was only one cat? 

Get estoppel letters from them verifying their rent amounts, the remaining terms on their leases (assuming they even have leases) and any security deposits being held. 

For your stove I would at least pull it out and check, verify the issues.

Look up a few on Youtube so you know what they're telling you is accurate (always verify).

I've worked with investors flipping,rehabs & new construction for over 20 verifiable yrs!

There is absolutely no way you can verify the cash rent payments, not even when the seller presents you with verified receipts.  

I pull a credit report (MySmartMove) and call to verify all references.