Impossible with just the basic inductive principle of tokamaks, yes. Some years back I learned that you can keep the current going with microwaves, not sure about recent progress. You can also approximate steady state by reversing polarity of the warp - ehm - magnetic field regularly.

Yes, stellarators are different.

The current ramp in the central solenoid is used to set the plasma rotation direction. Theoretically this is more like Alternating Current, since there is not a fundamental reason the central solenoid couldn't ramp back down (and then proceed to ramp up in the opposite polarization). The existing plasma would need to be cooled and removed first, or some similar mechanism of stabilizing the torus again in the opposite direction.

I look at it as a large optimization problem at this point. Each part of the machine is workable but not yet sufficiently optimized to achieve profitable operation.