If starting from scratch, doing standard drywall is fine. Make sure you have carpet floor (no concrete!!!). And drywall ceiling is recommended (once again, no concrete!!)
Second, make sure you have your electrician run one or more DEDICATED 20A circuits to the basement using 10awg ROMEX where your equipment is located. Two or three 20A circuits is fine, but if you only have one that's a minimum.
Third, your main router does not have to be in the basement, but make sure you run dedicated hard-wire ethernet cable to the location where your nvidia shield / streamers are located. Ethernet cable is always superior to wireless WI-FI.
Erik has excellent suggestions with acoustic treatment as well. Make sure the room is NOT square. Rectangle rooms are best with the front speakers on a short wall.
Fourth, running
speaker wire 14/4 gauge CL 2 in wall is really not the optimum solution. 14/4 wire is basically junk anyways and you'll end up wanted to replace it because it will become a limiting factor. In the end, it's up to you, but I never recommend limiting yourself to a type of wire.