I guess this is Twitch's rule in question:
http://www.twitch.tv/p/rules-of-conduct
"Non-gaming, non-music content: Video games, board games, card games, fantasy sports, LARP games, and acceptable music content (see above). And while it's nice to take a break and just chill out with your audience, please ensure that your channel's primary focus is on gaming or music creation."
In my interpretation, having a non-gamening related segment segment should be fine, as long as the focus of Gametrailers is on gaming in general.
(case in point: RocketBeans, who partnered their channel with Twitch, while this rule is in place)
No idea, if Kyle's stand-up will be gaming related. And having a game in the background might actually suit his program, so maybe he would want to do it anyway.