Move momma cat's litter box in the bedroom if that's where the kittens are sleeping, they should not be separated from her at night or at all until they are over 8 weeks old. My bitches (momma cats) usually shredded their own newspaper, I just threw whole sheets in the box. For the first few weeks, she will clean up after them while they are still in the birthing box. If they are jumping out of the birthing box then they need a litter box w/shredded newspaper nearby or you can buy the recycled stuff to put in there. Can you clear the floor of the closet for the box? Do not put their food/water next to the litter or they won't use the litter and probably will not eat or drink. When they see Momma cat using the litter they will learn from her example, which is why it's important to have the box where they can see it and have ready access. The sides should be low enough so they can get in/out easily or they'll find someplace else. I wouldn't crate them unless it was the size of a large dog crate, to give them room for sleeping at one end and a small box at the other--BUT, do not separate them from momma at this early stage! I raised two kittens that were separated about 2 weeks too early (allegedly 8 weeks, but I have my doubts), and have had problems that continue into adulthood because they lacked those last two weeks w/the mother. I advise against separating them at night, also; it places undue stress on both the kittens and the mother. Especially if this is her first litter, she needs to have access to them 24/7, access to plenty of high-protein food and fresh water until they're totally weaned.
The dogs will have to adjust for a month or so. It's good they can be socialized to each other, but the kittens take priority here. Our three had to adjust whilst the kittens were completing their training, and grown up enough not to appear as toys to our dogs.
BTW, good luck! Feel free to post back if you have questions. And if you don't want more of this in a few months, you'd better have momma spayed after this batch is weaned. Consult your vet for timing.