1. As a king, and even though he's gone out of his way to convince us that, ten years later, he still hates the more uptight aspect of the life, I would imagine that the calling of the kingdom is more important to him than the beginnings of the calling process. In fact, I'd imagine that at the urging of his advisors, under threat of political collapse for Ferelden, that he would endure it as long as he can. Also, they didn't vanish into thin air. They scattered and went underground (literally), but did not simply go poof.
2. I don't think Leliana actually knew where the warden was. I recall her saying that she would redouble her efforts to locate the warden if you advised it. If you do, she locates the warden (unless the warden previously romanced Leliana. In which case, I think it's more about location specifics and less about the "why)
Also, I think it could be considered that the blight, since it is a form of plague, will sour anything it touches, it would be best to conduct research in areas that don't have it. Is it resistant? How or why? Is there a reason blight doesn't linger there? Can it be connected to curing the blight elsewhere?
3. An immediate threat is easier to challenge than a concept of "end of the world." I commend the Warens efforts to save the thousands upon thousands of people who are about to die in the short term because of some ancient evil and manufactured magic. The warden mentions a lead, a breakthrough and a steady hunt. Closer than anyone else has come to curing it, why stop? No, she owes it to the world to keep going.
4. Fiona has lost the blight within her. It was miraculously purged from her, as she mentions multiple times. I don't know if she genuinely doesn't know, but I have my theories as to why it happened. No, she can't hear the calling. She is immune to the blight, and thus the magic associated with it. It's more than just losing the symptoms or entering remission. It REVERSED.
The cure would help wardens reaching the end of their limited lifespan. Can they cure the entire underground of darkspawn with the cure? No, blight exists. But merely, warden status would be temporary or not measured strictly in blood. It would be reintroduced during times of blight to slay the arch demon.
5. Leliana feels that becoming divine is greater than secular romance. Honestly, I don't blame her. That's something relatable and something that has definitely happened many times in reality. Bringing freedom to an oppressed people? Power to change the world, for better or worse? To have a chance to save the Chantry, the banner under which the last ten years of her life has been spent in sacrificial devotion? And where has the warden been? Where will the warden BE? No, this is more important.
6. I think the rank is a courtesy. Does one ever stop being a Marine? It is not so far fetched to consider her a warden commander even if she is not acting as one. She is a senior members regardless, and with a title like Hero of Ferelden.... Well, no one will dispute her worthiness as a warden anyway. I definitely don't.