Are all preprints on arXiv public?
Or is there actually a private unlisted preprint queue?
From behavior I've observed, I'm guessing maybe authors have the ability to hide papers and send private invites for select peer-review?
No they do not.
In fact, authors cannot even delete submitted papers after they have officially appeared in the (approximately) daily cycle. You can update your paper with a new version (which happens frequently) or mark it as withdrawn (which happens rarely). But in either case all the old versions remain available.
Postings go up once a day as a batch, so you could wait 24ish hours to see your paper appear, longer if your posting is just before the weekend.