Thank you!
You can display it within a block (not full page) if you want, but when we first set it up this way we noticed that scrolling messages within a page that also scrolls is a nightmare on mobile devices and not so good on desktop. Also note that Jot Messenger is not entirely full-screen - it still shows site header and navigation (side menu in Decorous).
I guess you may be referring to messages in popup tabs that show anywhere on site? Yes that would be great and is planned.
As for notifications - you should receive browser push notifications. If you aren't receiving them - try to clear cache, turn off notifications for your site in browser and visit again to receive notifications authorisation request.
The red notifications alert bubble in user menu will soon receive a significant update to make it more "real-time". :)