Is it possible to play against a "virgin" Leela Chess engine to see how many games it takes it to beat me? That is, can I train Leela Chess by playing against it myself?
Not an expert but you can play against a "virgin" Leela by using one of the younger nets. For example in the first Leela test (the so-called training run) the strongest net was 11248. If you play against, e.g., net 00010, it'll be much weaker.
However, you can't train Leela by playing against it. You can play against the trained product, but you can't train Leela yourself. That's because Leela is supposed to learn entirely by self-play (that's why its full name is Leela Chess Zero - "Zero" for zero human knowledge).
Yes, you could in principle set up Leela such that it learns by playing against you. It would require some programming skills and a lot of time to play games.
Currently Leela has been trained on 300.000.000 games. You should expect to have to play at least a couple of thousand games against it to see an improvement.
Also, Leela doesn't learn anything if it always loses. So you either can't start from zero, or you'd have to massively handicap yourself such that you regularly lose even against the initial random play.