1a: As said before, Leliana was killed in a place with magical healing properties, so she was saved from death by either the magical healing of the Ashes or perhaps even the Guardian.
1b: A case of gameplay-and-story segregation. While you cut her head off in-game, storywise she could have only appeared mortally wounded and/or dead, so the Warden left without discovering she had survived. (She already survived a knife to the gut in Leliana's Song, after all).
1c: You can't defile the Ashes without taking a pinch of them first to cure Arl Eamon, so it's entirely within the realm of possibility that the magic of the uncorrupted Ashes kept her alive.
2. Asunder is part of the Bioware canon, not our canon. Although this does raise questions about Cole's presence in Inquisition, which probably will need to be addressed since without Wynne, how did the events of Asunder play out or did they even happen?
3a. It's explained that the body was another mage and too burnt to be identifiable, so when people assumed it was him, Anders took the chance to bail as he never really wanted to become a Warden anyway. With both the Wardens and the Templars thinking he was dead, he could now live as a free man.
3b: He explicitly refers to Karl as the first man he'd been sexually involved with. Keep in mind that some people spend years as heterosexual before discovering they are either gay or bisexual, so this is not exactly uncommon. It is also possible to be bisexual and still have a preference for one gender over the other.
4a: Cullen lampshades in DA2 that the Templar recruits and the Circle Mages tend to gossip "more than a weaving circle". Since we know that the Templars rarely tell the Mages anything, no wonder they began to come up with their own explanations for him leaving, rather than just being transferred to Kirkwall.
4b: Cullen was a Knight-Lieutenant in Origins and was promoted to Knight-Captain in DA2. He was never a Knight-Commander, in the first game that was Greagoir.