(1) The Verizon DSL modem/router CAN NOT access the different sub nets. That is, 192.168.2.1 and 192.168.1.1 have to be routed to each other through the gateway of your Linksys router. Now to stop this and have a extention of your DSL router, you need to turn off DHCP for your Linksys router. WARNING: this will in effect turn off the ability for you to go to the Linksys router web page (you can fix it by doing a hard reset (hold the rest button for five seconds on the router)).
Now the most important part. Remove the Ethernet cord going from the Internet port of the DSL to a regular port from the Linksys. With this configuaration the DSL will now give you IP address and DNS entries. You can use the DSL routers configuartion page to do all of your port or routing needs.
(2) The other need way is for you to add persistent routes in a command prompt. Example: If your IP address is 192.168.1.100 then your gateway is 192.168.1.1 and you need to route the second gateway to your computer at 192.168.1.1. so you would bring up a command prompt and type
route add 192.168.2.0 mask 255.255.255.0 192.168.1.1 -p
This will add a persistent route for all incoming trafic from any IP address from the 2.x network (you have a 192.168.1.1 and a 192.168.2.1 gateway so they need to communicate).
(1) I urge you to just turn off the DHCP on your Linksys, then plug the cord to one of the four ports, not the Internet port. This will basicly extend your DSL router and the wireless. If you do this to the network, it will look like you plugged in directly to the DSL, and it should work just fine.