Your basic concept is OK. You connect each pc to the router, but you must use normal ethernet cable, not cross-over.
I think you are slightly confusing the roles of routers and hubs - for these purposes a router is a more intelligent hub.
You do not need a Gigabyte network card; only buy one if there is no price premium over a normal interface card since your router cannot work gigabyte ethernet; you would also need different cable (Cat 5e).
You should have no trouble with the Win ME setup - the retailer should have installed the necesary cab files in a sub-directory on the hard disk.
As well as anti-virus, you would be advised to run a firewall on both computers. Win XP has a firewall with very limited capability, but ME has none. They will not conflict with the router's firewall.