Will this bracket work with an aftermarket bumper cover?

This bracket is designed to mount at factory attachment points. If your aftermarket bumper uses the same mounting holes, you're good. If it has different hole spacing, you'll need to verify fitment or modify the bracket mounting pattern.

Do I need to buy a matching right-side bracket?

Yes — bumpers mount symmetrically. You'll need both left and right brackets for a complete installation. Order them separately or as a pair.

What fasteners do I need to bolt this down?

Fastener specs and torque values are in your vehicle's service manual. Bring that manual to the job — don't guess on bolt sizing or tightness.