What you packed sounds fine, though they will provide towels. Places I've been allow you to have your alcohol conditioner at certain times if you are on certain status, every hospital has different rules. Some will let you have strings and shoelaces, some won't. Some will let you have jewelry, some won't.

You won't be able to literally lock yourself in a room anywhere as that would be a safety hazard. But you may be able to stay in your room away from people. I was in one place where I could and another place that decided I spent too much time there and locked me out. Another that didn't allow patients to be in their rooms during the day unless they had a medical reason to be. So it depends on where you go. But honestly I'd advise against it. I know you're anxious and I have social anxiety too but it looks better on you and you get more out of it if you participate in groups and stuff.

Good luck, I hope they can stabilize you and get you on a proper treatment plan.