Dayly, that means one puzzle per day? And how many of them in total?

You can play more than once in the practice mode. It turns on after you complete the daily puzzle.

There's almost 20k entries in my human gene database. You can choose any of them as your guess, but the "gene of the day" is chosen from a subset that has decent 3D structure coverage (meaning, not falling back to alphafold2 for visualization).