1: At first I was thinking something for his own room, like a chest to contain his valuables (and giving him the only key to it), or a type of scrapbook to house all the pictures of the eventual guests that would come by now (since he has gone on to mention many different guests in the past and he seems to enjoy reminiscing).
But since it seems like you are looking for clothing options specifically...maybe a sort of tool belt to help with his work around the hotel? Or a type of satchel which can hold X amount of items as he travels the hotel? I guess a goofy/cute option would be a type of plush toy or teddy bear to hang at his waist (though I don't think they were around back then). Only other thing I can think of is a small instrument like a Harmonica to hang around his neck or something.
3: I guess I would tell him, "Take it one day at a time", or something like that. Even if it's an option that does absolutely nothing and he doesn't really believe it (might even be a little inappropriate considering how long he was stuck in that room for), it'd still be nice reminding him of that every so often.
As for what I'd like to do for him? Brush his fur of course! Or like, have him teach me how to handle brushing or washing fur since I can imagine quite a lot of guests coming in would have it. And seeing how the Player is a total noob regarding the Supernatural, who better to teach?
2. I would never send him out there (and never have), but if I had to come up with another reason as to why people would? Players may intend to sent him out there, and then sort of repay it later by doing/agreeing to something that the Player might not want to do. Sort of like a comprise of "do this for me and I will pay it back later". Have no idea if other people thought so, but it's one that came to mind as I was typing the other stuff, so I thought I'd include it here.