Belt drive systems are still an essential mechanical design component, both for power transmission and motion control, and certainly still covered in university classes these days.
Yes. You'll study the shit out of old school flat belts because V-belts, serpentine and cogged belts require more math that nobody wants to grade. Maybe you'll do one problem calculating how tensioner placement affects the force needed to make a cogged belt skip teeth.
The 4kW laser cutter I operate for work has (toothed) belts in it, so presumably the drive systems are covered under mechanical engineering?