He's the Last Dragonborn, a death like that would be an insult to the title. All Dragonborn through history have done great things and are rememberd, I will be very annoyed if they do another generic 'dissapeared in time' explanation as to what happened to him.
Not quite, there are several 'Dragonborn' in Sovngarde. If the game follows like every Elder Scroll to date, the hero will simply vanish after saving the world from Alduin.
Not every Dragonborn have done great things, besides Reman Cyrodiil and Tiber Septim.
The player character is rarely remembered past the plot story line.
The Last Dragonborn can be male or female, can be any race also. How would you continue that on, chances are the next game could be hundreds of years later. Historical events become myth and truth becomes muddled.
There would probably be a book that mentions the Dragonborn fighting Alduin, but unlikely much past that.