Cables that are rated for fire are not because the signals running through the cable will cause the cables to catch on fire but it is where the cable is physically routed to prevent that cable from being burnt. These are typically used in plenum spaces in building.
As far as the cable category, you want at least CAT-5e cable. This cable is capable of 1 Gbps bandwidth. CAT-6 would only be recommended if your were running 10 Gbps+ neworks. We're not quite there yet at the consumer level. However, if you can get CAT-6 cable for at the same or nearly the same price as CAT-5e there shouldn't be any reason not to.
One important note. Gigabit requires that all four of the twisted pairs of wires in the Ethernet cable are used. If you plan on running land-line phone service (sort of becoming a dinosaur technology today), you want to run an additional Ethernet or dedicated phone cable instead of trying to use a single Ethernet cable run for both services. It was typical in the 90's when a home was offered with Ethernet pre-wired they used the cable runs for both. This limited the Ethernet to 100 Mbps.