Can students see old course announcements that were previously imported?

Yes, students are able to see the previous announcements if they have been imported over into the current site.

Can they still see announcements if the tab is hidden?

Yes, they can only access the announcement only from the notification under the Dashboard via the Global/Recent Activity Stream view. They cannot access the announcements page directly using the breadcrumb navigation at the top of the screen after they click on the notification from the global activity stream.