On the last point, my experience is in CS, but it's worked similarly in other departments where I know PhD students and/or profs. When a professor at the places I've been hires a research assistant on grant money, the grant account is charged stipend + tuition (plus overhead, but that's usually already been taken out up front). It might be a reduced tuition, e.g. at state universities it's common to give an out-of-state tuition waiver so the grant is only charged the in-state graduate tuition rate, even for out-of-state students. But that's often still in the $10-20k range.

If the student isn't grant-funded, then yes, the prof isn't expected to pay them anything: the department will cover their tuition, and they'll work as a TA to earn their stipend. But at R1 universities it's nowadays expected that profs will fund at least a substantial proportion of their PhD students each year on RAships paid out of grant money.

The difference is not always obvious from the student perspective, though: when departments say they guarantee funding for 3 years, they just mean that somehow the student will be funded each year, not that the department will pick up the funding each year. The goal is to get the profs to pay for as much as possible out of grants, with departmental internal funding as a backstop.