Know about all of those quotes and such. Dragon Steel and Valaryian steel have similar qualities for sure and are intertwined but not the same in my eyes.
Supposedly the Lannisters had a Valarian steel sword can't remember the name of it, I think it was Lan the Clever's.
Tywin melted down Ned's and made Widow's Wail for Joffery and one for Jamie that he gave Brienne (Oathkeeper?). Not sure if ole Joff was buried with his or if it was given to Tommen. Not that Tom would ever have a need for it.
Think he was talking about a family sword of their own. Think it is this one... Brightroar, the ancestral sword of House Lannister, lost when King Tommen II Lannister of the Rock sailed to Valyria and never returned.
Lip reading to find out what Lyanna said to Ned came up with this... Jon Sand name could be Jaehaerys Targaryen. Jaehaerys is indeed a name befitting a Targaryen baby. Williams writes that Jaehaerys actually even has ties to the Night’s Watch. “Jaehaerys I was the 4th Targaryen king to sit the Iron Throne. He was known as ‘the Wise,’ ‘The Conciliator,’ and ‘the Old King,’ Williams writes. “His long rule was prosperous and he was aided by his sister-wife Alysanne, who convinced Jaehaerys to expand the Night's Watch and granted them the land now known as the New Gift. http://www.refinery29.com/2016/06/115346/jon-snow-true-name
He's not necessarily a "Sand". Some bastards are named for where they were born, some for one of the parents, and others for where they are raised. "Snow" is perfectly acceptable, as is "Waters" (Crownlands).
No not necessarily but if he was a normal bastard born in Dorne and didn't get a family name he wouldn't likely have been named Snow. With Ned taking ownership that changes things.