Do I need fasteners, or does this come with bolts?

Fasteners aren't included. You'll use the hardware that's already in the factory mounting holes on your frame—no new bolts needed unless yours are stripped or missing.

Is there a right-side bracket, or just this one?

This is the left-side bracket only. The right side is sold separately. If your bumper needs support on both sides, you'll need to order both.

Can I install this myself, or do I need to remove the whole bumper?

You'll need to remove the bumper cover to access the mounting points. Once it's off, the bracket bolts to the frame at the factory rocker and frame attachment locations—standard wrench work from there.