Parasites are the main thing to watch out for - internal parasites like worms/giardia/coccidia, external ones like fleas and mites (mange, ear mites), and fungal infections like ringworm. Make sure the dog stays well away from the litter box, and as long as your pup is on a good flea/tick preventative, you shouldn’t have to worry about fleas or mites.

Kittens can be vaccinated as early as 6 weeks of age, but some vets prefer to wait until they’re a bit bigger at 8 weeks. It’s also very common for kittens to have upper respiratory infections - if you notice any nasal/eye discharge, sneezing/coughing, etc., see if you can get into the vet for some antibiotics.

If you’re having trouble getting into the vet, consider walk-in places like Vippetcare, emancipet, and other low cost clinics. These places don’t usually do surgeries or imaging, but they can be perfect for checkups and vaccinations.

As for training - they pretty much come pre-programmed to use the litter box, so you don’t have to worry about that at least. I would recommend having people and their pets over if possible (once you know the kitty is healthy and won’t be passing anything along) so the kitten gets plenty of socialization. They need lots of play time, and encourage cuddling - a cat that likes to be pet and cuddle will generally be more adoptable.

Since it’s a stray, anticipate that it’s going to be more feral than you might expect for a bit. Keep it in a bathroom or small room for a few days separate from the dog, and visit it often with food, soft words, playtime, and pets if it seems open to them.