I don't believe that Longclaw is "Lightbringer" in the traditional sense from the old times. I believe that Azor Ahai's sword becomes Lightbringer because he wields it. Azor Ahai would predate steel in any form by hundreds if not thousands of years, as only the more recent Starks had steel swords. The older tombs had brass and iron weapons that had completely rotted away. Hell, even Ice(Ned Stark's Valyrian Steel greatsword that was melted down into Widow's Wail and Oathkeeper for Lannisters) was not the traditional Stark Greatsword ICE and had only been around for a couple hundred years at the time the novels take place.

Jon is the battleground on many layers. He's both fire and ice as a Targaryen and Stark. Not purely either.

I don't believe there's anything special about Longclaw beyond the fact that it is a valyrian steel sword. Now, Dawn, on the other hand.

Starting to buy into the theory that there is a definite connection between the Kings of Winter(Starks) and the White Walkers. Not sure if that connection is adversarial or friendly at this point. But the term KOW seems to put them on the same side. At least in relation to Fire anyway....