Internal parasites can be transmitted to other animals, vet will need to do a stool sample and if the cat has worms you'll need to medicate it.
Feline Leukemia is the biggest problem with strays. Some cats carry it and it doesn't do a thing to them. Some cats can live with cats that are carrying it and never catch it. Some vets can be pretty ruthless when it comes to this stuff. I've never allowed any of my cats to be tested for it because they were all living together for a period of time before they went to the vets so
Have the vet check the strays ears, if they're crummy tell the vet you'll take care of it yourself. You want to use stuff called Zymox, get it from 1800petmeds. You don't need to go through the B/S of treating for mites and crap like that, just squirt the Zymox in and it'll clear everything up, no need to swab.
FYI, if you do bring it home and your cat has never HAD another cat in the house before, your old cat could very likely be fucking FURIOUS.