No desktop required. My guess(tm) is that you were looking at a system that uses Microsloth ICS (Internet Connection Sharing) to provide the wireless router function from a desktop. A PC running ICS is used in place of a $30 router. This is common where the DSL modem is on a PCI card or uses USB, instead of the more versatile ethernet. I haven't seen it with cable modems, but it is possible.
In my never humble opinion, this is a lousy way to share a connection. It would be better to replace the USB or PCI DSL modem with an ethernet DSL modem and to use an ethernet router, instead of the PC with ICS.
Yes, that's sorta the way it's normally done. However, you still need a router with NAT (network address translation) to share the connection. If you just connect a WAP (wireless access point) directly to the cable modem, you will get exactly one wireless connection and no more. You need the NAT services to share the connection, which is why you need a router somewhere in the system. Therefore, you need either a wireless router box, or as I prefer, seperate boxes for the ethernet router, and the WAP.
No problem. I prefer to answer the easy questions and leave the tough ones for others to deal with.
My cable modem ISP is set up to recognize a single MAC address. The installing technician will want to connect one computer directly to the cable modem and do the configuration. If there's a desktop, of course that will be used. If there's a router already connected, it might work, since the tech will type in the IP address of the registration page. A laptop would certainly work just as well as a desktop.
The technician might want to connect the computer directly to the modem and not use the router.
If the technician installs with the router, then no particular machine will be required after that. The router will be registered. If you want to change the registered device, that is possible. You can also "clone" the registered device MAC address onto most routers so that they can be used on a cable modem that has a single PC registered on the system.
Some cable systems don't have MAC restrictions, some do. Some can be "trained" to use a new MAC address, some require registration. The technicians seem to be cable TV techs for the most part.