Hi Alex,
You enroll once you’re here, I think that’s what the majority of university students do. You can enroll locally, they will get you set up with a student visa if you’re on a tourist visa. The process is very straight forward and simple, it seems like you’re already familiar with most of the major points. No worries about getting kicked out of anything because of visa problems, as long as you don’t overstay your visa you’ll be fine. And make sure you register when you arrive or this could happen to you, hahaha.