Sorry for the delay, my internet connection was down
This should explain everything
As the other posters already have said, your switch is getting ONE IP address from your ISP. That you can use for ONE computer or for a router.
The router stays with the ISP's IP address and builds up a local network where you can attach the computers.
I have my old desktop computer under my desk without use, I will set it up as a headless router
Maybe it would be working with the switch if you would set up one of the computers as a router, with two NICs or wireless cards, and attach the other computers to your computer acting as a software router.
I can't, I use the switch for this
I don't understand how can you attach the computers of your dad and brother directly to the cable modem.
Yes (continued below)
Can't your Access Point run a DHCP server of its own?
I have tried this but the computers with the private IP addresses don't get internet this way
Nevertheless you should be able to run a DHCP server from your Wireless Access Point (and essentially do NAT), so that all computers in your network are assigned private IP addresses.