Is there a way for me to set up a specific time to play a specific opponent in real time online? For example, is there a site where I can arrange to play, say, Bill Smythe at 8:00 PM CST on 13 November?
Alex Relyea
When I want to play someone at a specific time online, I just talk/email/message the person and get their handle name and just agree to be online at the agreed upon time. Lichess, and I’m sure most other servers do, allow you to offer a match to a specific opponent.
With a full-blown audience watching (via the forums or otherwise, I assume). You must want to refute some of my absurd opening lines. 
Bill Smythe
You can do this on Chess.com, ICC and/or any site that allows live chess between two individuals.