Feeds: solution with most frequent/real time/push notification update of feeds (e.g. for twitter tweets)?
It looks like Twitter's "Streaming API" is your solution. However, I would look closely at their Pre-Launch Checklist before going down that route. They have 10+ things that you should comply with before becoming a stream or push consumer of their API.

Maybe a solution that would fit you is to develop a twitter stream client (that would be your server), and then to have your application users connect to your server via node.js, since node.js provides a "persistent" connection suitable for chat rooms and push-like notifications.

This is an interesting (and probably doable) suggestion, but not something that you would find pre-packaged in a Drupal module.