As mentioned earlier, the fire det loop may be detecting a fire with the gear up but may not detect the same fire once the gear is lowered because the flames are directed away from the loop due to the changed airflow. The extinguishing of the light does not signify the extinguishing of the fire.
Regardless of whether I thought any fire, including a cabin fire, was extinguished, I would get a shuffle on to land at the nearest available. Any QRH fire drill that permits the flight to continue I would take umbridge with - you can't always determine if the fire is out, will stay out and hasn't already spread to somewhere undetectable. In my books, a fire will always result in an immediate landing; it's the one emergency that has no extra time beyond that for the quickest landing possible for troubleshooting. You get the thing on the ground ASAP and run what checks you can along the way.
I can see the point that Rat5 and 16024 are making, but if the fire is spreading so rapidly that these other systems are compromised in such a short time, then the airframe is going to unzip pretty quickly - better to skid off the end than disintegrate in flight.