Q: Front bumper cover/support --- do I need it?

Them's fightin' words!
The front bumper would most likely protect you when you hit someone because you dive bombed a corner. If you were really worried about our driving, you would move the honeycomb to the sides of your car.
If you think you are a good enough driver, get rid of both bumper cover and support as both add weight, and you won't be going fast enough long enough for aerodynamics to matter.

Point taken.
Perhaps I should just screw pillows to the side of the car.

Our bumper cover is gone (mostly because we broke it taking it off the first time, and it would just be trying to fall off anyway). We kept the support, however (which, in our case, is a giant styrofoam block).

If you can get cash for it, whatever. If you can't, I'd probably run with it.

The other important thing to consider is whether your bumper cover has a net benefit or net loss on cooling ability and airflow. Usually they're a net loss, but occasionally some bright spot got it right (probably by accident).

I believe in trimming all the weight you can.  Our car ran with no roof, no windshield and no trunk.  Still we kept both bumpers.  Sure you only use the front bumper when you hit someone, but I remember night driving at Nelson Ledges and watching a car in front of me go into a spin and then then the "oh crap" moment when I smacked it and stopped it's rotation.  I thought for sure I was doomed.  I drove on from the hit wondering how much damage the car had sustained.  The plastic GM bumper was barely scratched.