The network being uses is private, the "public" network is the internet. The IP range you would be getting is 192.x.x.x or 10.x.x.x correct?
You can setup a subnet off his network to separate the data from the others that use this connection but you need a router capable of that. This can help you get started https://supportforums.cisco.com/thread/2101560
Using wifi will complicate things a bit as you need to get a wife access point to connect to instead of just wiring up an ethernet cable to the common router.
That IP range is correct. If I can't do it with the devices I currently have (linksys dd-wrt) then I don't want to break the bank to make it happen. If I were to purchase something to make it work I would buying something like the Buffalo wireless dual band converter.
I'm not positive I understand what you are trying to suggest in that link. It seems like they are trying something more complex than what I want.
I want to receive his wireless signal on my router and create my own SSID to broadcast out. In theory I feel like this should work with only two routers. I'm thinking there may be something in the middle that is required to make it actually work?