Well being Lyanna and Rhaegar's son has been the most popular theory for years- to the point I would actually be upset if Martin made it that obvious. Plus, he looks the most like Ned so I still believe he is Ned's son.
I think his mother is Ashara Dayne of Starfall, who Ned was smitten with. Sister of legendary knight Ser Arthur Dayne. And he could be Azhor Azhai. His sword Lightbringer will likely be Ser Arthur's greatsword Dawn. That is what I'd prefer but let's see what GRRM gives us.
As for being dead. Possible. But highly unlikely. GRRM wouldn't leave it a cliffhanger then, would kill him there itself. Plus, I don't think he's dying without knowing who his mother is. Maybe he warged into Ghost, maybe Melisandre revives him like Thoros used to revive Ser Beric. Maybe it wasn't Jon Snow getting stabbed at all. Or *shudder* maybe he becomes a White Walker. Anything is possible but fairly sure we'll see more of Jon Snow. I've always considered him to be the "Ice" of "Ice and Fire" after all.
Don't see him sitting on the Iron throne but will possibly inherit Winterfell at some point. I guess the Night's Watch will disband and the Wall will come down eventually, or the realm splits back into seven kingdoms.