Taking in a friendly stray, what to get her checked for at the vet?

There is a stray cat that's been hanging around my house for a long time that we feed. It's incredibly friendly, loves to get pet, doesn't mind being picked up, etc. My mother has been wanting a cat for a while so we decided maybe she could adopt this one. But, before she does, I want to take the stray to the vet and get it checked out. This cat is not the picture of health. It doesn't have many teeth and walks with a slight wobble, as though maybe it had an ear infection a long time ago and it messed with its balance. No idea how old it is. So I want to know if there's anything seriously wrong with it besides the obvious.

I was wondering, what kind of things should I ask for the vet to check for? I know checking for FIV/feline leukemia and seeing if she/he is neutered. I also want them to see if it has any parasites, but that requires a stool sample and I have no idea where to get one because the cat lives outside. Is there another way for them to check that?

Also! The cat has some matted fur on one side and does not smell pleasant. Could the vet groom him/her for me as well? 

My mom lives with my grandmother while she takes care of her and I don't want to send a super disease ridden feline into a house with a delicate older woman (which is why I worry about the parasites in particular since I think that is contagious to humans?) Just want to get this animal a relatively clean bill of health before I hand her over. Thanks for any advice!

The vet can get a sample from her anus, so do check for parasites. She may need to be treated for fleas, ticks, worms and ear mites. Her wobble may be from malnutrition, so if there is any way you can increase her food, that might help.

The vet should also check for a microchip and determine if she is spayed.

Bless you for helping her. Fingers crossed your mom and she hit it off. It sounds like this kitty could use a break.