Show only unread content; remove item once read or option to remove items
You can filter nodes exactly this way (a node filter called timestamp I think). 

I'm not sure what the best way would be to solve the "x" button thing. Maybe add an extra binary field to the contenttype which you set via ajax (triggering a function you defined via "page callback" in hook_menu()) when clicking the "x". Then you have to add this field to your views filter as well.

Edit:

Instead of an extra CCK field you could also insert data into the history-table (uid, nid, timestamp) of drupal and pretend the user has already viewed the node. This way you would only need first views filter.