I requested a book on 5/23.  The requested was accepted on 5/28. 

The Label Print Date, Date Mailed and Estimated Arrival are all still showing as 'awaiting member'.  If the other member doesn't confirm that the book was mailed soon, will it flip over to the next person in the FIFO line and when will that happen?

I've never noticed this situation before (though I'm sure it must have happened and I just never noticed).  The book is part of a trilogy for a deployed guy and I've already received the other two books... just waiting on this one. 

Yes it will flip to the next FIFO eventually.  I think it's 2 days past when they said they'd mail. If a mail date isn't showing but they have accepted, I think that means they marked Can Mail Today, but not as sure on that.

They get 48 hrs past the "will mail by" date to mark the book as mailed.  Well, it could be a bit longer, things flip over in the system at like 3am.

Once that time is past and the book is not marked mailed, it will roll over to the next requestor.  If the first person actually does send you the book, you do not have to give them the credit since they didn't mark the book as mailed in time to stop the request from rolling over.

If you're not in a hurry for the book, you can always cancel the request if it rolls over to the next person just to see if the first person really did mail the book.

It sounds like they chose the I can Mail in 2 Days (there is no 1 day option) which would have been 5/31 (I don't think it counts the day the book was accepted because it gives 2 full days-I could be wrong). Anyway you have 2 days after that to mark it mailed.  So if the mail by date was 5/30 then it would time out tonight and go to the next person. If the mail by date was 5/31 then it would time out tomorrow.