Wireless and LAN will not run at the same time. Once WAN connects, LAN drops out. There are ways to overcome this but really, you either use one or the other interface, since routing can be a problem, and requires additional configuration.
This answer talks about these problems
In terms of power, is the Pi still on after you plug in the USB Wireless? My Pi reboots some (most) times when I do this, but I have the very first editions. This got resolved with newer PCB.
Try and connect the Pi over HDMI and see what happens when you are directly connected to the Console. Does it reboot? Maybe it crashes? Thats the best way to debug.