wireless is not good for FP games, lag is to great.
You will need to add a router or switch, depending on the functions of your box, (if you only have 1 port).
As far as the PC you are going to build. You will proably NOT need to add any nic card. Most all MB these day come pre equipted with 1 and some have 2 gig ports.
So what you need to know is does the cablebox have a built in router? or just a single lan port. If built in router is their more than 1 lan port? Is the built in router have a firewall? If firewall and single port you will only need a switch to expand your system. For add security you could use a router.
Your IP address could give you a clue, if 192.168 or 10.10 subnets you have a router. If not you will need a router.