Now that the question rate is dropping off.
Should we have a beta new question feed in chat?
I think we should. Many main chatrooms have at least one. On Blender we have several. One for our main site, one for meta and one for the blender.org rss feed. We have bots post the latest questions on main and meta and a ticker for the other feeds. As you noted since the question rate is gradually going down for now, I don't think this will be distracting or really annoy anyone.
It has several advantages, the main being that those who lurk in chat can see new posts and possibly have a discussion before tackling it. If the messages get annoying, these can easily be disabled temporarily.
Another use is to watch other related sites for possible questions to migrate. We shouldn't worry about migrating anything yet but just putting the idea out there for the future.
Yes - as long as it doesn't prevent folks from actually enjoying and relaxing in chat. One of the hallmarks of a good, healthy Q&A site is a relatively short amount of time before we deliver what we promise to users:
The sooner that can happen, the better the user's overall experience, and the more likely they'll become an evangelist for the site. As chat is generally active, it does make perfect sense to feed in new questions at this point.
If we reach a place where folks just aren't amenable to the noise any longer, you can always start a second room for people that just want to swap pet gifs while waiting for more questions to come in :)
Additionally, as more tag specific rooms start forming (cats, dogs, fish, ferrets, etc) - I'd encourage those respective room owners to pull the corresponding tag feed into the room.
Disclaimer: Skippy is alerted to new questions by sensing vibrations in the ground, and hops to answer them immediately. Everyone else should use chat feeds.