Did Tohbo Mott ever teach Gendry the secrets of reworking Valyrian steel?
Interesting question.
Who or what were the Stormsingers? What could they do?
Another eastern magical discipline.
What happened to Ser Arthur Dayne's sword Dawn after Ned brought it back to Ashara?
Dawn remains at Starfall, until another Sword of the Morning shall arise.
In Valyria did they work Dragonbone into Valyrian steel?
No.
Did Tyrion ever finish reading the book concerning dragons that he borrowed from Winterfell?
Yes.