Do you have a question or is this supposed to be a tutorial?
A network can only have a single router in any subnet. Using multiple routers is possible in a network, but they each need to be in their own subnet. Otherwise cascade routers LAN - LAN and disable DHCP on the downstream routers, which effectively turns them into APs and switches if you use the remaining three Ethernet ports.
Double NATing routers isn't difficult, but unless you know what you are doing or follow exact instructions you get a poorly or non functioning network. I have three routers double NATed behing my Actiontec and they all work fine and I get 85/40 when using Ethernet. (Still waiting for the upload boost to take effect.)
I have a question. Unfortunately in order to ask my question, I have to have a lengthy description of my setup. Basically, I have a second Linksys router in "router" operating mode with NAT disabled connected to the Verizon router, and I have a computer which is in a different subnet (192.168.2.x) behind the Linksys router. This computer can communicate with computers behind Verizon router in subnet (192.168.1.x), but cannot reach internet. This is a simplified version of my question, full details are in the original post.
If I setup the Linksys router in "gateway" operating mode, which means with NAT enabled, then both intranet and internet works, but there is no easy way to setup port forwarding for 10 compueters in 192.168.2.x network to communicate with 10 computers in 192.168.1.x network.
If I setup the Linksys router in a LAN to LAN configuration with Verizon routers, but this way all computers are in the same subnet, I want them to be in different subnet for access control and things like that.