My avatar driver that @mattias mentioned could certainly be part of a solution. It depends on whether you want the notification text to always visible on the dashboard or to have an indicator that there are notifications through something like an avatar image, then for the user to click on that image to display them, like notifications here in the Community. I can't remember if I ever really got a pop-up dashboard properly working when clicking on a notification dot, or just an external link... It could be something I could look at doing if you are interested.
The display of the rolling text is definitely possible. My go to would be outputting the text to a file and then using an html file to display the text as an iFrame in a tile. I did something similar with a prototype of an EPG tile here. There are also examples on the CSS thread for scrolling text on a tile here.
In terms of the logic, that's probably a hard one to wrap up neatly in an app, but the retention of notifications until they have been actioned / dismissed or some other logic deems them obsolete, that is something I have been interested in for a while, and am not sure I have seen something like that yet. Can't commit to building it, I have enough half finished projects at the moment 
Although, I would probably derail it slightly by wanting to extend it to dismissing notifications sent to my phone as well.... Using Tasker.
