Will these mount at the factory fog light locations without drilling or cutting?

These are designed for factory mounting points. Test-fit before finalizing installation to confirm alignment with your specific vehicle's bumper opening.

Do I need to run new wiring to the switch, or does it tap into existing fog light circuits?

Wiring configuration is not specified in the product data. Have your install shop verify whether the integrated switch connects to existing factory harnesses or requires standalone wiring.

Can I use these on their own, or do they require the vehicle's headlights to be on?

The integrated switch gives you independent control—you can run them anytime. Check your local traffic laws regarding fog light usage.