Yes, there are definitely people who study this and can get P.h.ds in Music Theory I believe, but obviously not all of it is relevant to someone just trying to learn basic piano skills.

I'd say that for making learning and playing the piano easier you should learn about chord structures and progressions, so that you can essentially "free-style" if that makes sense. You'll find that a lot of songs follow the same progressions (you've probably seen the four-chord song on YouTube).

I'd also take a look at ear training so that you can have a better relative pitch (identifying what each note sounds like).

I'm not sure exactly where I'd "draw the line" but honestly if you just spent a few hours with the stuff above and had some charts handy while you learned it would be helpful. As a fan of music you might find it interesting as I do and want to go further with it. There's no way that I wouldn't help you, it just might not be necessary for the level of skill you're trying to obtain.

I know there's a lot of online resources that are free, but I personally haven't used any so couldn't recommend them myself. I have heard good things about teoria, but just look around to find something you like -- I'm sure there's even mobile apps for it.