Yeah that will be Push Notifications usually based on something ready in the app or a 24 hour delay. I've used Urban Airship before where you can send Push to users whenever you want.
Yep. Although, you still have many options in this plugin I've made.
(Rich Media Features)
You have Local Notifications (Offline Background Events), Rich Media, Messaging and the regular Push Notifications. There should be more of those but for those who want them have to experiment it themselves.