|I was wondering if I could connect, via wire, the G and N routers together and configure them so that the wireless PCs had their choice of the N or G router, whichever worked better.|
This should work just fine. If it were me, I'd setup the downstream router to offer the wireless network and nothing else. Which is to say, have DHCP shut off on it and probably the firewall as well. Let clients connecting to that router get their DHCP from the first router (via the wire of course).
Just remember to put the second one on a different channel. Which is to say, the first router on the 3'd floor should be channel 1 and the second router on the 1st floor should be on channel 11 to prevent any interference.
You will want to use a crossover cable between routers if at all possible. While not absolutely necessary, as both routers should be auto MDIX capable, it's still preferable to use a crossover when connecting like devices. You will connect from LAN port to LAN port and when configuring the second (downstream) router, you'll want to give it an IP in the same subnet as the rest of your network, but use an IP outside of the DHCP scope so as to avoid duplicate IP's. Use the exact same subnet mask and your DNS and Gateway addresses should point at the IP of the upstream router.
One consideration. Just because a router is N, does not make it's signal stronger. However, with two in operation, and the second one of the two on the first floor, all wireless clients within the house will have no problem with signal strength.