Will these bulbs work with my factory fog light housing?

Yes. These are H10 bulbs sized for OEM F-150 and Mark LT fog light assemblies. They drop straight into the existing socket without modification.

Do I need to remove the entire fog light assembly to swap these in?

Not necessarily. Most folks access the bulbs from behind the housing without pulling the whole assembly. Check your owner's manual or service manual for the exact access point on your truck.

Are these street-legal?

H10 is a standard DOT bulb type. Your truck's fog lights were originally wired and approved for this spec, so legality is not affected by swapping to these smoke-tinted versions.