Re: New Kittens!
They were sent home awfully early, I would never send a kitten home before 10 weeks, preferably 12 weeks if I were a breeder. Shelters obviously can't have kittens that long without fosters so I expect 8 weeks from them, but never 7. They may just not have had a good litter box habit yet.
Kittens like to potty on absorbent things so i might move the "quarantine" to the bathroom and take up any bathroom rugs. this way the litter box will be the most absorbent thing in the room. Another thing to remember is that a toilet is the toilet because it smells like the toilet. If they have peed on the bed a few times it needs SERIOUS cleaning with an enzymatic cleaner (it must be enzymatic, even things marketed as pet cleaners are useless on pee unless it contains enzymes), soak the area very well. It would help if you first cleaned it with water to flush out as much of the urine as possible and then used the enzymatic cleaner to rid the mattress of the leftovers that you can't smell but your cat can. The hardest thing to do when it comes to mattresses is drying it, I would stand the mattress up and aim a fan at the spot, I might end up using a hair dryer as well. You need to make sure it is clean before you make the bed up or else it's probably mold or something.
Also, are you using the same litter that was being used where the cat came from? Some cats are picky or just don't like new things. So watch when he uses the box to see if he appears to get a lot of litter stuck in his toes or refuses to dig before and after going potty.
the bed is probably just attractive so even when you let him out of the bathroom again I'd shut bedroom doors for a while unless he is supervised by someone. Make sure he doesn't dig on the bed or pace and circle or show any signs that he wants to go. If he starts that scoop him up and bring him to the litter box. Keeping the bed made (if you don't) might also discourage this, if he is unable to dig up blankets and cover his waste with the blankets he might not go there. If you want just keep him off the bed always. But it is usually just a matter of habit, once he's used the litter box for a few weeks never fail he should be good. Also make sure he knows where the box is located and maybe get an extra one for another location. If it continues and he hasn't been to the vet yet I'd bring him in, I suppose he should be having a check up soon anyways since you just brought him home.