Is there a way to tell if the person you're requesting a book from is running a deal? Will a little box pop up or something?

Nope. Often the person will have put a notice at the top of their bookshelf in that header area, but not always.

You can always just ask. Some people consider this rude others do not. I ask. The worst they can do is say no. If they are rude then they are someone I wouldn't want to trade with anyway.

You can do a forum search by Nickname.  That should tell you about recent deals.