Pretty much any relay you see tagged with Arduino will handle automotive 12V accessories just fine. Whether it will survive the vibrations and possible temp changes is TBD.
Use a USB car adapter. That line will already be fused.
Are you sure your factory fog light button is momentary and the on/off somehow controlled by a computer? Or is it a latching button under there that you've concluded wrongly about?
Are you wanting separate controls of fog and this accessory light so that either or both can be on, or will they always be on together?
Please clarify and confirm so we're not all working with bad info.