As heavyd said, the address for the modem needs to be on a different subnet than the one controlled by your router.
Just to clarify, the subnet mask shows which part of the IP address is the subnet. A subnet mask of 255.255.255.0 means the first three numbers in the IP address are the subnet.
So in your setup, both the router and the modem have the subnet 10.0.0.x. When you enter the address 10.0.0.1, the router sees that address is on the subnet it controls and never sends packets for that address out on its WAN port.
So, as heavyd said, the solution is to change the router's IP address to any private IP address that doesn't begin with 10.0.0.
For example, you could change your router's IP address to 10.3.3.3 and leave its subnet mask at 255.255.255.0. In that case the router's subnet would be 10.3.3.x. Then when you enter the address 10.0.0.1, the router will see that address isn't on the subnet it controls and will send packets for that address onward to its WAN port.
No. You can create your own WiFi that's routed to the existing WiFi. You can bridge wired clients to the existing WiFi. But bridging wireless clients will not work.
We tend to think of WiFi as working just like wired Ethernet. But unfortunately, it's really not. It has its own rules and this is one case where it's just different.
The short version of the problem is this: An access point will not accept packets with origin hardware addresses that are not paired with the access point. Since some clients won't be able to reach the existing WiFi network, they won't pair with it, and those access point will reject bridged/repeated packets from them.
What might work, and I stress might, is two bridges back-to-back, connected by wired Ethernet. The specifications say it shouldn't work, but in practice it seems to. Apparently, while the WiFi hardware isn't capable of proxy registration, the DD-WRT software is. Bridging from wired to wireless is done in software in these routers, and DD-WRT is smart enough to proxy register a software-bridged packet. (It knows it has to do the WiFi magic on behalf of its wired clients, since they obviously won't be able to do it themselves.)
For this setup, you connect two wireless routers, LAN-to-LAN. You configure the WiFi on one as a station on the existing WiFi. You configure the WiFi on the other as an access point on a new network. You will probalby need to use a different SSID though, so people will specifically have to choose to connect to your network.
Theoretically, you could use the same SSID and encryption, if any. This would allow clients to automatically connect to the strongest network. You will have two issues. First, you have to make sure your two routers don't connect to each other wirelessly. That obviously won't work. Second, clients will typically show a temporary disruption of service when they switch to and from your access point because the existing network won't know where to find them. MAC learning takes up to a minute. You might get complaints if you 'steal' people who momentarily came into range of your access point and experienced service disruptions as they switch to and from it.
Update: The way back-to-back bridges work is by a form of NAT. The bridge doesn't proxy register its clients but instead re-writes their source hardware addresses.
Best Answer
To use DD-WRT as a router, you have to connect to the primary router via the WAN port. Using a LAN port will use the bridge functionality.
If you properly connect to the primary router via WAN, here is what you need to do to function properly.
Assume Primary router is 10.0.0.1/24
Assign (or DHCP) an IP address to second router (on primary), say 10.0.0.2/24. This makes second router another device on the 10.0.0.0/24 subnet and it can talk to all devices on that subnet. Set its gateway to be 10.0.0.1
Use a different subnet on the second router and setup in NAT mode. say 10.0.1.0/24 and give it an address 10.0.1.1/24. Enable DHCP on this router as well, since you want it to give out addresses on its local domain.
This allows all devices on the second router to talk to each other. And for all non-local addresses to send it via the uplink for the primary router to handle.
On the primary router, add a new route.
destination 10.0.1.0/24 -> 10.0.0.2
All responses coming back to the 10.0.1.0/24 address range now will be forwarded to the secondary router.