You can use CAT5e or CAT6, but just make sure you use plenum rated cable (this is for fire safety). You do not need a switch, just an all-in-one router/modem (which may even be wireless), which is sometimes provided by the internet service provider, and normally has 4 ports. I would just do wireless, it is easier. I hate running cables through walls. If something isn't wireless compatible, you can buy a wireless adapter pretty cheaply. Installation is pretty easy to as long as you get a decent brand. I have had very bad luck with IOGear.
If you needed more ports than provided by the modem/router, and didn't want to go wireless, you can get a small switch pretty cheap, in which you just connect that to the modem/router as if it were a computer (meaning in the same ports). It will pass through pretty easily.
Just set the file server up on the LAN. Connect everything to the router/modem and you will be fine. They will get private addresses via DHCP from the router/modem.
Gaming service is just through the internet.
The security camera just needs the camera hooked up to the server. If you run the cable through the walls that it fine. You might need to go get some RJ-45 wall mounts so it doesn't look crappy. To run the cables into the RJ-45 wall plates, you will need to strip the cable's coat, then strip the 8 pairs of cables down to copper and stick them in the back of the RJ45 wall mount plate with a screw driver or another tool that I can't think of the name of.
Found a picture of what it could look like! The router would be a router/modem. You need the modem to receive internet access. The router is for DHCP private addressing, so that multiple people can use the same internet connection. If you need to, put the other router between the modem/router and the computers, so that you can have extra ports.
If I missed anything, or was a little vague in any area, please comment...