Does the wired XBox 360 controller work with Surface 2

There is no way. Microsoft made it so the Play and Charge Kit only receives power. I believe there is some other data that is sent to the controller as well that someone decoded but there is no data sent to the computer from the controller. I will try to dig up a link.

On the other hand, you could just buy a wired controller, those are about as expensive as the wireless receiver, work on both PC and Xbox/Xbox360, and you'd have an extra controller for a friend.

I have seen this several times myself as well. Is your Xbox360 hooked to your TV with an HDMI cable? If so, on several occasions I have tried to turn on my Xbox without having the TV on previous to the attempt and my controller does the same thing. My hypothesis is that someone or something accidentally bumped your controllers home button and it is trying to turn the console on without the TV being on, which the console will not do for some reason, thus causing your Xbox controller to blink even though the console is in fact turned off.(Note that this may be possible even with the normal cable hook-ups, I just don't use mine so I can't say for certain.)