If you already have game working on LAN, the TCP/IP version of it is no different at all. There may be some issues that need to be taken care of for packetloss, latency and disconnections but the network components and usage are the same.

In regard to routers, that is something you can't do much about. Loads of people posess internet connections that are not capable of acting as server, therefore you can either put up a real game server that all games send packets through (as any mmorpg does), or be happy that some people can create server to host games to their friends