I am also experiencing the same issues. Surface Pro 1 with Touch Cover 1, upgraded to 8.1.
The keyboard will sometimes become unresponsive, which can usually be resolved by detaching and reattaching the keyboard multiple times. Sometimes when the keyboard has become unresponsive, the Windows hardware icon on the device stops working. At that time it is sometimes possible to fix the issue by tapping the modifier keys on the keyboard (I think that Alt may be the culprit, but I tap Ctrl, Alt, Shift, Windows one at a time to try to "unstick" them). Even more, sometimes the device does not only become unresponsive, but a single letter will stick, repeating the character over and over until the keyboard is detached. Curiously, at the point the keyboard is detached, the word completes with the remainder of the keys that had been pressed on the keyboard AFTER the stuck key began to repeat.
And then of course there are the sudden reboots, which seem to be related to the keyboard. They occur especially often when the keyboard is moving. For example, I frequently picked up the device (with keyboard attached) and turned it around to view the Serial Number, this caused the device to reboot twice.
I hope we can get a statement on whether this is a widespread issue that they can/will address in a software update, or if those who are affected should schedule their device to be serviced.