Assuming your data looks like this:

Then I'd recommend having a form in the Registrations table with a linked field to the Slots table, where you set the records that can be selected to be filtered like so:

And this is how it would look in use, and a link to the form can be found here

---
For the reminders, if you're alright with sending out one email per slot, then that's really simple. You'd have an automation that triggers once a day, and its action would be to look for any slots that are reserved for that day, and then use a repeating group to send out emails as needed:

Link to base
If you wanted one email to be sent per person, regardless of how many slots they've booked, then that gets more involved. You'd either use a script helper or create a new table that contained all the people that have reserved, and then use that table to build your automation