Which is a better choice for shooting classroom teaching, a DSLR or a shoulder-mounted video camera**?
Shoulder Mounted video camera? 

I think that's a little over the top, A decent 1080p or even 4k video camera (Canon, Sony, Panasonic) would give you a decent image, your main thing would be audio, you want something that would give you decent audio, a camera with a XLR input and a shotgun mic to start with and then maybe mic yourself up using a wireless kit.

Obvisouly this would all depend on budgets, time, amount of editing you want to do, maybe if you want to live stream.

Other things like decent tripod, recording media, etc will affect your budget.