The same exact thing happened to me a couple of weeks ago, then maybe a week later, I saw the notification that the update to Samsung One UI 2.0 with Android 10 was available.
So I have it, but let me warn you that it's REALLY buggy, rspecially when you try to open links from Google searches. It seems like nothing is happening, and if/when you go to another app after getting frustrated, maybe ten+ seconds later, whatever browser you have as the default will pop open all of a sudden.
This doesn't happen all of the time, but often enough to be frustrating. But that's what you get when you use Betas!
No, you have to wait for the invite banner to appear on the first page of the Members App.
The S10 series beta is full, the Note 10 series beta should be starting soon.