Skype has a master overlay(which just means when it's called, it's registered priority is important, so you see it above all windows) that may interfere with the newer VAC authentication. When someone calls you and you're doing something else, you get a big picture with all the info.
Take this for instance:
I've had this same problem, but I've never been banned for it. It kept constantly kicking me out of the servers until I turned off Skype.
This was on an old PC. When I built my new PC, I didn't have this interference problem anymore when I ran Skype while playing CS:GO. The fact that it's fixed to CS:GO might hint something, as this was right around the time of CS:GO initial launch.
This isn't really a solution as much as it is for hope. I'm not sure why you're having those issues exactly, but if you continue to have issues in CS:GO after your cool off, please message me and I'll spend time investigating.