It depends on the team....and how they want to use him. Starters typically go 4-5 innings depending on pitch count. Relievers may throw more than once....again depending on the team, how many pitchers they have and the pitch count. Most teams will schedule their pitchers....both starters and relievers so that coaches know when to expect them to throw. Obviously being a starter gets you more opportunities...and usually a few more innings. It's much easier to convince a coach to come when he knows you'll throw 4 or 5 innings than if you're only going to throw 2. As far as the hotels....again, it depends on the team. If you're a starter, you're usually just obligated to show up for the game you're throwing, but I've never heard of any that cared if you wanted to stick around. We had several kids that did....just to hang out and be a part of the team. If you're a reliever, and scheduled to throw the third game of the weekend, you may get a chance earlier if you're there.....and possibly be seen by a coach who happened to be there to see someone else. I guess it's up to you and your son on whether the extra expense of staying extra nights will be worth it.
As far as mid week camps/showcases....as long as your son is ready to go on the weekend, I doubt his team will care what he does during the week. Again, starter/reliever may determine how much he can throw other than the weekend