Hi.
Pianos are sort of mechanical masterpieces. To me anyway.
That said I have rejected (surprizes even me

) even some free ones.
IME, while a passable tuning and repairs can be performed by a DIYer, a pro can do it 10 times better.
The best way to diagnose the dropping key problem is to carefully study every part of a working mechanism, from the key to the hammer, to figure out what is each parts contribution to the whole, and then try to figure out which part fails in its function in the faulty one.
Fortunately You have only one key to worry about (for now

), some that I've seen has had half of the mechanisms in a plastic bag.
As for the tuning wrench, a mom and pop store will be able to order You one, it's doubtful that they'll carry one in their inventory. Unless it's an acoustic piano store of course.
Regards
Sam