- If he's inside of a wall, can he see/hear what's happening in an adjacent room?
Sure, just add the perception mod of the stone walls remaining thickness to his check.
- Can he cast (while he's inside of the wall) into an adjacent room?
Nope, he cant see the area so no line of sight. Plus hows he meant to execute the verbal component when talking into dirt. Its like having a rag in your mouth and trying to talk clearly.
- Can he emerge(what exactly that means?), cast a spell and then go back inside the wall?
Sure, if his movement permits or he has the appropriate feats. The walls like air so it doesnt effect what he can normally do.