I would also like to see when a missing assignment is submitted, the headline of the notification should make it clear that a missing assignment has come in

This would be really helpful- even a badge-type icon with an number by it (by assignments??) to show you that things had been submitted.

I really like this idea.  But I do not want an e-mail notice but rather a counter next to the red/yellow/green box that indicates some activity in the assignment has taken place, such as, a submission or discussion has taken place.

This is an awesome idea!

Same thing for getting notifications when somebody comments on a news or participates in a discussion

We are a Blackbaud/Education Edge client, and we recently opened up OnCampus to our entire school.   There has been great positive feedback, but one of the largest (overwhelming) requests coming in only 2 weeks after launching is to enable an email notification to teachers when students complete their assignments online, whether it be a regular assignment, an assessment or a graded discussion.   Though FAWeb isn't as interactive as OnCampus, this is one feature that all of our teachers had in FAWeb, and they are already missing it sorely now they have moved to OnCampus.