Many people tend to keep their rabbits in a smaller area for the first few days, and gradually introduce them to more space after bringing them home, for a smoother transition. Ultimately, it’s personal decision. Do you know if he was fully litter trained with them, or did he still make accidents around their home? To my knowledge, litter training rabbits will not necessarily be spot-on unless they’re neutered/spayed. The pooping amongst the area can potentially be territorial droppings, which is mainly seen in rabbits who are not neutered/spayed. If he’s neutered, maybe try blocking off the corners or placing another litter box in those corners, to see if he will use it. My bunny has always smushed his face against the cage/stood up when I’m about to let him out for the day, so I’m unsure if he will ever stop. It could be from being a younger bunny and/or pure excitement, but I’m not sure. Congratulations on the new bun, and I hope you figure everything out soon!

Do you know if he’s neutered? Cause that could cause some issues as well.
Bunnies tend to do their business in a set corner that they favor. So honestly, the best solution is to stick 2 litter boxes in those corners and fill them with hay, and odds are he’ll be fine.

If he’s standing up when you go near the cage and putting his face near the door then it’s obviously because he wants to get out. Why would he stop? If he didn’t do that at his previous home it’s probably because he was bored/depressed. Or because they had the cage open during the middle of the day.

You can’t keep him confined to his cage until he is litter trained. You can put a pen around his cage and put tarp down to make clean up easier.