what role does the router play for connecting to the internet?

A router is a device that forwards data packets between computer networks, creating an overlay internetwork. Usually, routers simply pass data between computers and the outside network, but many are more intelligent than that, including services such as firewalling, IP forwarding, IP masquerading, etc.

You could have two routers, but it's not really necessary, unless you have a large amount of computers or you need more than one ISP, or you have special network traffic needs. For example, you may want to separate your office's network (which shouldn't be accessible from the outside) from, say, your company's webserver and mailserver (which must be accessible from the outside).

If you just want to connect several machines to a single network, you just need some network switches and a single router.

Routers allow connection of multiple computers to a single internet connection.

Routers assign IP address to the computers that are connected to them.

Yes you can have more than one router providing they are connected correctly.