Any time diarrhea has gone on for than a couple days, you need to get that checked by a vet. It could be a virus, bacterial infection, parasites (if he goes outside), eating something he shouldn't have, etc.
You can give him some plain yogurt twice daily 1 tbsp to help the diarrhea. It helps restore natural good bacteria in the digestive tract and does a great job firming stool.
Switch from wet food to dry food if you can. If he doesn't like dry food mix some infant rice cereal in his wet food.
You need to be careful of dehydration as well. You can give him a tsp a couple times a day of unflavored pedialyte. You can use a medicine dropper to administer this. Give him 1 cc a few times a day. This is equal to about one tsp.
It is hard to tell if a cat is running a temperature without taking it with a thermometer. Cats naturally warmer than us, so they always feel a bit warm to our touch. A normal temp range is 100.5 to 102. If you can take his temp that would be best to know for sure.
It's possible. The most common thing that the new kitten could have passed along to your cat is worms, if they share a litter box at all.
Almost all kittens and pups are born with them so worming them upon adoption is extremely important.
Also a new kitten causes change which causes stress. Stress alone can cause diarrhea and stomach upset in cats.