You have to have a current medical certificate prior to solo.
The FAA gives you an option of using your social security number on your license or they can issue you a unique number for the purpose of your certificate.
When I got my private (25 years ago) I tried not to have my SS# as my license # but they wouldn't allow it.
As far as the medical, you are right, I had to have it before solo, I didn't recall it as I think I went and got it the first week that I started lessons and it was a non issue to me.