Separate the two cats until you can get a stool sample from each one and you know definitely who produced what. Put them in separate rooms or put one of them in a separate room and let them each have their own box, food and water. Leave them separated until both produce a sample for you. Label them by name and take them to your vet for microscopic analysis.
Little kittens often come with intestinal parasites. Older cats often get diarrhea from stress or GI or metabolic conditions common to older cats. It could be either one of them or both having the soft stool. You cannot guess. You must find out for sure.
As far as the kitten's behavior is concerned, I would not call it aggressive. The kitten means no harm, but is doing what kittens do. They want to play every second that they are not eating or sleeping. The behavior is normal. Your other cat will adjust to it somewhat, but also you can give him some peace each day by putting the kitten away for awhile in a separate room.
Did you have the kitten spayed also? If not, that will tend to calm them down somewhat and is better for their health.
Good luck and if you have further questions, please don't hesitate to write again.
Cat box problems