I always thought of it as a ligature for "et" (like ffi is a ligature for ffi) rather than a letter. But I suppose the sz ligature (ß) eventually became a letter, so there is precedent.
Not really, depending on who you ask. In the 90s, the RAE declared that ‘ll’ and ‘ch’ were no longer letters, but they’re still taught as letters in many places. ‘rr’ was never a letter in its own right, but many people consider it to be, as a kind of parallel to ‘ll’ and ‘ch’.
It's called es-zett (S Z), and I believe it originally was a joined long s and a z, like ſ𝔷. (There doesn't appear to be a fraktur long s in unicode, but I include the fraktur z to show where the shape comes from.) I believe these days it is typically written as "ss" when the ß character is not available.
Edit: The name points to sz, but it's possible that it replaced both ss and sz. I'm not an expert.