IP Addresses are only needed if you are using the TCP/IP network protocol for your devices to talk to each other. If you are going to give your computers access to the Internet then you have to use TCP/IP as that is the network protocol that the Internet is using. However if you do not need Internet access you can use IPX/SPX or NETBUI instead. (This will depend on your OS, as newer Microsoft OSs don’t support either of these protocols.)
Depends on what applications you are running and what services they need access to. Other protocols include AppleTalk, SNA, x.25, etc.