Does another non-apple router can be added to this existing wireless network with the devices that I already have?
Apple designed their "extend" function as a proprietary feature that will only work with other Apple routers, so it is highly unlikely that another device by another manufacturer would be able to wirelessly extend the signal of an AirPort Extreme.
However, if you plan to extend the network by connecting the new extender back to the "main" AirPort using a permanent, wired Ethernet cable connection, then virtually any device from another manufacturer should be able to do what you want.
Over the years, I have tried a number of devices from Netgear, D-Link, Linksys, etc and none of them were compatible with an Apple router to wirelessly extend an Apple network. But, I suppose anything is possible and there might be a device that will do what you want. Stay tuned to see if another user who has a different experience might respond.
I'm not sure where you are located, but Apple routers are still available as new, refurbished and used device from Ebay and Amazon. You might want to check that out.
Finally, a note to mention that if you decide to try another Apple device, the new AirPort must receive a stronger signal from the "main" router than it does from another AirPort that might already be extending the network for things to work correctly. Another way to say the same thing........the new extender must be located closer to the "main" AirPort than the other AirPort Express that you have. The reason for this is that an extender must connect directly to the "main" AirPort.....not to another AirPort.
