She definitely said "A...." So Jon's real name starts with an A, and since it'll be something Targaryen-like, it must be either Aegon, Aerys, or Aemon imo.
Yup, all he care at this point is recruiting people against the White Walkers. If he (along with everyone in Westeros) does learn about his parents and his rightful heir to the throne, i could see Danerys and Jon marrying (since it was heavily foreshadowed by Daenerys) or at least make an alliance
Her list was: Joffrey, Cersei, Tywin, Walder Frey, Meryn Trant, Red Woman, Beric Dondarrion, Thoros of Myr, Ilyn Payne, The Mountain, and The Hound. Quote from like Season 4.
I got two theories for Daenerys. Either she'll be corrupted by the Throne and become Mad resulting in her death. Or, she'll realize the true danger (White Walkers), form an alliance/marry with Jon, and defeating the Night's King.
Imo, i don't think she's going to die. I think she's going to sit on the Iron Throne, while Jon rules the North (or possibly dying after serving the Long Night), and Tyrion to whatever he likes. I doubt Tyrion will end up being on the Throne with Jamie by his side.