First of all dd2p_3 seems to be the devolo powerline device. These devices do not assign IP addresses (they don't run a DHCP server). So your Router 2 will try to receive a DHCP address via dd2p_3 -> dd2p_1 connection from your modem.
However depending on the modem it might be limited to one local IP address. You did not include details about your internet connection. But "modems" typically do just bridge the connection to your ISP. So finally your LAN devices get IP addresses from your ISP directly.
I live in Switzerland and here this setup is common for cable operators (Cablecom) where the ISP provides a modem which just bridges the LAN and a machine attached to the modem gets an IP address directly from the ISP DHCP server. Unfortunately it's also common that such ISP do provide only one single IP address to their clients. So the first router connected would get the public IP assigned while the second one does not get one.
The solution would be to change your setup slightly using only one router:
modem -> Router 1 -dd2p_1 --> dd2p_2 -> Switch 1 --> PC1
| |-> PC2
| \-> PC3
|
\-> dd2p_3 -> Switch 2 --> Server PC
|-> PS3
\-> XBOX 360
In this case your Router 1 gets the public IP from the ISP and internally distributes local IP (usually 192.168.1.0/24) to your clients. Router 1 would in this case perform NAT translation hiding all your internal stations behind one single IP.
If you would like to separate LAN segments you might also use a more complex setup:
modem -> Router 1 -dd2p_1 --> dd2p_2 -> Router 2 --> PC1
| |-> PC2
| \-> PC3
|
\-> dd2p_3 -> Router 3 --> Server PC
|-> PS3
\-> XBOX 360
This setup is more complex as you have to use independent networks and if PC1/2/3 have to access the server it requires special routing configuration too. That's why I would recommend the setup above for simplicity (all hosts managed by single router).
Of course it would also be possible that simply the connection between dd2p_1 and dd2p_3 is not working. This is quite common if you connect powerline devices to different power rails in an apartment. However reading your description I would bet that your "modem" just hands out one single IP which is picked up by the first router so the second one does not get an address then even if the connection on powerline is fine.