I'm assuming they no longer had their personal items in the house.

It must be considered as ABANDON Property and there's  a procedure for every state.Don't get sucker punched on this - - if you dispose of it improperly, he has a case against you.In Calif:send certified Notice of Abandoned Property, itemizing items left on the property.He has 18 days to reclaim and pay for storagebeyond that, if it's worth less that $700 do as you pleaseOTHERWISE, you must auction it off, and post the proceeds with the State.Follow the procedure, even if the certified letter is returned - - that's your proof of due diligence and will dismiss any case he might bring later.

There is no line item for improvements like new appliances.

During the annual inspection the only item for correction was a bush touching the house. 

Would the exact costs of some of the items included in the closing costs be unknown at the time of agreeing on the purchase price?

It's a small part to give up in exchange for other items.

If I agree with their finding, I sign off on it.To note the original condition, I use the abbreviation C & F (clean and functional) on every line item.

Contractor's estimate is $800 for labor to repair and re-paint both closets and the bedroom.The 3rd item, drywall/painting, is the one the tenant has the heartburn about.  

At that time, you will not be allowed to start depreciating the item, so you lose out on the deduction in full and will likely owe the IRS back taxes with interest and penalties. 

In other words, I had to earn $75,000 to buy a $50,000 item.