A link only they can book

A booking URL for one event type with a random hash where your username would be, like cal.com/d/7Hq2xKmN9pRtV3wYbC4sLf/board-prep. Anyone who has it can book that event without seeing your username.

A new link allows one booking. In its Link settings you can raise that number, or pick a date instead, and then the link takes any number of bookings until the end of that day.

No. Each link runs out one way, after a number of bookings or at the end of a date. If you need both kinds, add a second link.

It stops booking. Your list marks it Expired, with Usage limit reached or the date it ran out under it, and anyone who opens it gets a 404 page.

Yes. Add new link makes another, each with its own limit and its own count, so every guest can get a link of their own.

No, private links work on any event type. Hiding it keeps it off your profile page, so people find it through the link you send rather than by browsing.