First Time Foster, Deaf dog Marking (?)

Hello! I'm giving a lot of detail, my main concerns are bolded at the end.

Notes on the dog: he's 2 years old, 70 lbs (which is his healthy weight), and untrained (but not feral, he was surrendered by his original owner when he was about 1, when their financial situation changed). He's my first foster.

I picked up my first foster on Tuesday afternoon (11/13). He was described as housetrained and knowing "a few" hand signs, though talking to the surrendering adopters (they signed the surrender papers after 3 or 4 days, kept him until I could pick him up on day 13) they told me he had zero training, but was both crate-trained and house-trained. He spent a few days at an overnight doggy daycare before I picked him up.

I'm familiar with the basics of training a deaf dog. We've gotten a few sits successfully, and he's largely stopped jumping on me for attention. His previous foster used a knocking motion for "good boy," and I'm using some modified ASL for "sit."

About the marking/accidents. Wednesday morning, his first morning, I feed him and he marked the corner while I was eating breakfast. I interrupted him by moving him by the shoulder (aimed for quick without being painful) cleaned up, leashed him, took him for a walk. He had no issues for the rest of the day, despite me having a long work day (coming home for my lunch and dinner breaks to walk him).

Today, I walked him right away, before breakfast. Then we did a quick vet trip to have his paws checked (they were red and irritated). He has allergies but is fine. I also picked up a larger crate from the rescue, since the one I had on hand was just a bit too small. He was showered with treats during the check, but manhandled quite a bit since he is entirely untrained.

We came home, and he had an accident while I was setting up the crate. I was too slow to fully interrupt him, but I moved him to his pen while I cleaned up, then took him outside for a walk. Then I left for work (evening shift today, so I could make the vet). Came home for dinner, walked him. My roommate came home about an hour after I last walked him, mostly ignored him while they were making dinner. He peed on the door to the outside.

I know this is a lot of detail, and I know there are a lot of stressors that could make this genuine stress-related accidents. He has been moved around a lot. His paws are itchy/sore. I've reached out to the previous foster to see the exact details of his house-training (was he going on a matt? Did they have a doggy door he could use whenever, instead of waiting for a walk?).

My main concerns/questions:

– am I accidentally teaching him that peeing inside = fun walk time? He is *very* quick with patterns, incredibly intelligent, and very attention-seeking. I play with him during my meal breaks as well as walking him. I ignore him while I clean up his accidents.

– my roommate and I signed up for a house-trained foster. I would really, really, really hate to move him again, because that is almost definitely contributing to his issue, but we live in a rental and both work full-time–we just can't house-train a dog. Plus, my roommate hasn't signed up to be backup-foster-parent; they just like dogs. How long should I give him a chance to "prove" he's house-trained?

– experienced foster-parents: how common are accidents with newly arrived fosters?

I apologize if this is somewhere in the Wiki, I'm feeling rather anxious/urgent about it and didn't see anything specific when I skimmed it.