Does the LMS contain base content types specific to your solution? If so, are we able to edit them if need be, to add mandatory columns?

Yes, we use custom content types and as a rule they are registered on a list or web level, but most of them are hidden in order that site admins don’t break LMS functionality. You won't be able to see and modify them but for the rare exceptions. We do not support such process as you might meet issues during future upgrades.