It can be a number of things. First reading, it just sounded like they gave you a feral cat, with the loose stool coming from stress and chane in food. But after more reading, I would have suggested the vet trip like you did. Can be distemper, not common in cats, but deadly. Also, Heartworms. Less likely due to the age. Also, heart failur, also not common in kittens, or poison. Even kidney failur.
You really should not have just dropped off the kitten. You should have gotten an estimate in what they where planning in doing. They should do FIV/Leuk test, over all blood panel, and radiographs. Also less likely, but possible - rabies. Even head trama if it is feral and then ran head first into things.
Poor little kitten.
You should get a vet approved bill of health for ANY animal - dog, cat, bird, ferret, pig, horse, ect. Unless you are ready to take on 100% of anything that might happen I the next week that could have been provented from a simple vet check.
If your other cat is vaccinated and the kitten and cat did not get into fights that cut skin, it should be fine.
Also, remember an animal that is okayed by a vet one day, may turn a 180 due to underlining problems.