Being a cat momma myself I can pretty much guess what's going on IF IT'S NOT A HEALTH PROBLEM.
Poor guy doesn't like the situation, most likely is not too thrilled about the dog. He hints that with peeing and pooping on the dog bed.
What you can try is Feliway. Especially in the area where the dog bed is. It should make him more comfortable there....the cat that is
It does work, my Princess came from the shelter, 9years old, just dumped when we were there to get two cats. She is a long haired cat, had absolute NO hair on belly, down the hindlegs, all licked off. We thought it was an allergy first, tried cortisone, which stopped her licking for about two weeks, then she started again. So we got Feliway. It took one little container about 4 weeks and she has stopped licking herself bare, has now a full coat.
So I think it's worth a try. Get it from an online vet supply, huge price difference compared to PETCO.
If it's health related, which the vet will tell you, there are dry foods especially for UTI problems.
Less minerals. Another thing, the water. My cats get only bottled or filtered water.
And you can try Cosequin cat. Its not labeled for UTI problems but it coats the ways, urethra and urether so if he has crystals or stones, they pass easier. Tried it, kept a couple of mine well.
I found that when a cat has these problems, they keep going to the box over and over again, without producing much and will make sure you'll notice.
So my guess it's a behavioural problem, most likely related to the dog. Or the new situation. Or both.
Oh, if he has stones, he can be operated and will be fine after that, btw.