Do I need to run new wiring, or will these work with my truck's existing fog light circuit?

These are designed to work with your truck's factory wiring. No switch is included, so if your F-150 doesn't already have fog lights, you'll need to source or fabricate a control switch separately.

Will these fit if my truck doesn't have factory fog lights?

These mount at factory fog light locations. If your truck doesn't have the OEM fog light buckets, installation won't work without modification.

Are bulbs included, or do I need to buy those separately?

Bulbs are not included with these assemblies. You'll need to source the correct replacement bulbs for your F-150 model year.