Kittens can be neutered or spayed as soon as they turn 2 lbs. I would set up an appointment within the next week, no later, if I were you.
Our cats prefer having their own litter box, each, so I agree with the previous poster that you should get a second litter. The kitten may hate going in a dirty box, too, so keep them as clean as possible.
I find that Nature's Miracle (an enzyme cleaner) does wonders on cat accidents, so stock up! (at most pet stores)