I don't really like how Skippy is stuck in my inventory until I arbitrarily kill enough people and wait days for him to give me the quest to get rid of him. I think high Int or Tech should be able to hack him and bypass a lot of the wait time for this.
You have the quest or you still "wait". Because it require to be equiped if I'm right. If you keep it in your inventory, it will never trigger the quest (I my case, it's never too long, few in-game days at max).