You can get routers which use a SIM card for getting internet access. They're pretty cool. You just plug them into a wall outlet and bam, you've got your WiFi with internet access.

Plus you can even get them with integrated powerbank, so you really don't need to plug anything anymore.

Yes, some wireless routers can be used as repeaters.

Routers like that are really multi-purposes devices that handle routing, switching, and wireless. If you're only using a local network, you can still have the wifi network and all your devices communicate through that. You just won't have internet access.

With out the router/switch, the cable's usefullness is reduced - at least in the networking application we're talking about (the cable itself is incredibly usesful for many other things besides ethernet). You can connect (most) devices directly together for communication, they'll just self assign an IP address.

In less words, you can have a network with just a wireless router and no cable, but you can't have a network with only a cable and no router (well, you can't with 'normal people' devices)(also you don't need a router for a local network, only a switch - but the home versions are often combo devices and just called a router).