Do I need to remove the bumper cover to install this bracket?

Yes. The bracket mounts behind the cover, so you'll need to take off the cover first, secure the bracket at the factory attachment points, then reinstall the cover over it. Standard hand tools handle the job.

Is this the complete bumper assembly or just the bracket?

Just the bracket. This is the reinforcement support piece only—you'll need the bumper cover itself as a separate part.

What if I need both left and right brackets?

Order the right-side bracket separately. These are sold individually so you can build the exact configuration your truck needs.