Is it possible to fire an engine to slow down atmospheric reentry?

Yes, of course.  However, you would need as much fuel to bring the speed down from 17,000 miles per hour as it took to get up to that speed and you would have to boost that fuel into orbit, making the rocket even bigger leaving the ground.  That is why we take the free braking the atmosphere gives us.

I'm going to try here to help. Lets say you unwind several miles of tethered drag chute into the atmosphere. It will of course begin to catch air at some point and play out behind the craft. I guess then it's a race between slowing down and having it jam you into the atmosphere. But...I guess if you keep the craft skipping on top of the denser stuff while the chute does it's work, eventually the craft will begin to stall. So a controlled stall all the way into the atmosphere....?

Combine that with some retro  and I guess ...ah well...I've gotten off the question haven't I?

YES BUT
it would take almost as much fuel to slow down as you used to get up there. actually much more fule to get up because it would have to lift the unused fuel for the return

big problem for a mars return mission

space ships mostly coast but reenter at space speed

air friction slow down is the only way whe have

why do  Drag race automobiles need paraschutes at the end of the track.  Brakes are just not good enough to slow down form 250 mph.