Do I need fasteners or clamps to secure this to my exhaust?

The clamp itself mounts your tubing together. You'll need hose clamps or band clamps around the exterior of the sleeve to lock it in place once your tubing is seated inside.

Can I use this to join two existing exhaust pipes?

Yes, slide each pipe into opposite ends of the sleeve until they overlap in the middle. Both ends need to fit inside the coupler bore for a proper seal.

Will this work if my tubing has rust or old gasket material on it?

Clean the tubing ends with a wire brush or cloth before sliding them into the sleeve. Any debris trapped inside will prevent a tight seal.