Sure, the opponent would point it out, just like they would presumably point it out if you played any illegal move. In serious tournament games they would probably also stop the clock, call over the arbiter, and inform him or her that you made an illegal move so you can be penalized (e.g. under FIDE rules if you make an illegal move your opponent gets 2 extra minutes on the clock).
That doesn't change that it's an illegal move.
For sure. I didn’t realize moving into check was an illegal move in the sense that I’ve only played casually and the opponent (or myself) points it out.
Yeah, "illegal move" is just a synonym for "things you're not allowed to do". There's no separate category of moves that you aren't allowed to make, but that aren't considered illegal.