Experiencing the exact same issues with my new 2014 Sierra. Only picked it up yesterday. Not impressed.
OnStar Remotelink works to unlock/lock, but when I tried to start it this morning I got an error. Tried with both key fobs, and it did not start. Finally I went outside to try to start it with the key and experienced the 'click, click' issue. Dash was all lit up. I tried a few more times, got out, swore profusely, then remembered that the salesperson said something about key cycles. I figured maybe while I was messing with the remotelink I overdid it and had to lock/unlock a few times or something.
That didn't work either. I finally got it to attempt a turnover and got a 'Low Battery' warning on the display. So now it's plugged in. The temp here is -22C today. I never plugged in my old vehicle (the one I traded in for this one) and didn't have any issues.
Needless to say I'm not impressed so far, which is a shame considering the sales staff and everything up to this point has been top notch.
The internet tells me that it might be a frozen exhaust vent, or it might be a greasy selenoid. When it finally starts again, I'll be taking it directly to the dealership, because this is unacceptable.