Google generally approves almost any open source work. They're also really generous about approving even for-profit side projects as long as there are no conflicts and you're willing to wait for approval. I was cleared to work on a few side-project games while I was there.

Re: do-nothing tolerance, the dirty secret at Google is that there's too little meaty work to satisfy all the people hungry to take it on, and everything takes ages to do anyways because of all the red tape, so it's kinda hard to tell who's phoning it in deliberately and who's getting stuck in the constant muck of cross-team approval and bikeshedding bullshit when they take 4x as long as expected to deliver half of what was promised (which mostly doesn't have any impact anyways). And managers are (were?) rated at least in part based on how many people they manage, not what their teams produce, so dead weight doesn't matter to most managers, if they're trying to get bumped up to director level the last thing they want is the distraction of having to PIP and then fire someone when they could just keep them around as padding and say they hit their staffing target...