Ok, here's the best I can answer going off my own understanding.
1. This may not be an oversight; I warn you, there's some spoilers ahead regarding the book 'The Calling' and an important character in the DA Universe:
2. Same reason Leliana said the Warden and Champion's dissapearance 'couldn't be a coincedence' or Morrigan seems to have forgotten whatever important revelation she had about her mother at the end of Witch Hunt - the writers of the game wrote themselves into a corner, and decided to write themselves out of it at the expense of making the characters look stupid. They clearly didn't plan for this 'Calling cure' business back when writing the DA2 ending.
3. The Warden was written 'out' because we'll never be able to have them back, properly - back in DA:O, your character's personality was nuanced and decided by you depending on the situation. Now, in current DA, they will have to have a personality and a voice that may not match what players believe their characters would act like, nor what they recall their Wardens sounding like (if I recall there was something like 6 VAs per gender in DA:O - no chance of getting them all back in to voice even a cameo). A lot of players will be upset that their Warden is no longer 'theirs' because it'll be impossible to 'get them right' for many people. That being said, the Warden
is a popular character, and Bioware understands that if they survived the Blight, they wouldn't just 'dissapear'. So this 'Calling cure' was made to distract them and give a reason, no matter how flimsy, why the person who stopped the Blight is somehow unavailable to help the Inquisition save the world. The only way we'll see the Warden back is in a silent cameo, or as the player character (which is unlikley, considering they may be dead for a lot of players).
4. No, a Grey Warden who is cured of the taint cannot hear or sense any Darkspawn. This is evident in Fiona and the answer to question 1's behaviour. Grey Wardens are needed to kill Archdemons because, when an Archdemon dies, it's soul seeks out the nearest Darkspawn to possess - Darkspawn are souless creatures, so the Archdemon's soul fills it not unlike an empty vial. But a Grey Warden 'confuses' this process - the Archdemon's soul is tricked into being drawn into the Grey Warden themself as the nearest 'Darkspawn', due to the taint. But a Grey Warden is not an empty vial - they have souls. So when the two souls colide, they are destroyed; this is why a Grey Warden must be the one to kill an Archdemon, or it will just body hop whenever their current body is defeated.
5. If becoming Divine, Leliana's priorities probably change - she can't shirk the responsibility, and she realises she's probably the best hope for enacting real change so can't leave it to anyone else. When not becomming Divine then, she either plans to try and recruit the Warden into the Inquisition, or it's forgetful writing on the writer's part.
6. It's never specifically stated
when the Warden-Commander leaves Vigil's Keep; the closest we get is that they stay 'for a time' or 'a few years'. Technically, a decade is 'a few years', so they could leave any time after DA:I and still be consistant with the Epilogue. It's best to recall that the Epilogues themselves have, unfortunately, become non-canon for the most part - the writers of Dragon Age pretty much ignore them whenever they write new games, so don't take it as 100% fact.
Hope these answers have helped a bit.