if he pays it does not give him any sort of 'ownership' rights over the property, and he cannot remove them when he leaves.
put it all in writing and make him sign it.
if the door does not need replacing then you are under no obligation to replace just becasue he'd prefer it.
my dad once had a tenant who redecorated her living room simply because she wasnt keen on the wallpaper - it wasnt particularly nice, but it was clean and in good condition and therefore perfectly functional - she tried to demand my dad then cover the cost of all the paint, brushes and decorator she employed.
she was outraged and felt why should she have to pay for decorating HIS house ... turned out some idiot had told her to just do it and send the bill, utterly convinced this was acceptable... because of some garbled interpretation of the landlord being responsible for repairs and maintenance rule.
it wouldnt surprise me if someone has told this person that if they pay for enough repairs over time they somehow develop a share in the property.
its possible that they've heard about how sometimes, when a couple split up, when one is not the owner but has spent a lot of money on the property over a few years, they are entitled to some sort of a share...and got it all mixed up and think this will work.