Airport Express bringing down the entire network?
You might be having a situation in which you have a conflict because both devices are trying to act as DHCP servers, and perhaps serving addresses in the same range. 

Make sure your new Airport Express is configured in "Bridge" mode before connecting it to your network. I don't have an Airport here, and don't remember exactly what Apple calls the WiFi Router modes.  You want a mode in which the new Airport Express does NOT do DHCP or NAT routing, and just passes packets to and from your main Airport router. 

Get it configured in Airport Utility before trying to connect it to your network. You can temporarily connect it to your computer with an Ethernet cable. (Use the Airport's LAN port for this.) 

Then, you should connect an Ethernet cable from a LAN port of your main Airport to the WAN port of your new Airport Express.