I can't for the life of me find a way to tell Illustrator to save the files that I have currently open and reopen them after I restart. Am I missing an obvious setting?
That can't be done.
You can use Bridge to open several files at once (or just select them all in the Finder/Explorer).
But there's no way to specifically control how tabs are ordered.
Nice idea. Just not possible as far as I know. No Adobe app retains "sessions" like you describe.