Flemeth and Solas issue was deliberatly left unclear as a cliffhanger.
In my opinion it looks like Mythal (Flemeth) jumps into Solas' body. We know that for that to happen he had to be willing (he's talking about "paying the price).
It's unlear what happens to Fen'Harel if my opinion turns out to be the truth.
Why do I think so? Because there is not a single documented event or even a relation from somebody who did/witness the reverse process - absorbing the essence/soul from someone else killing that someone in the process.
- Archdemons jump into a different body, and their old body dies.
- Corypheus jumped to a different body, and his old body died.
- Flemeth jumps into bodies of her daughters, while her old body dies.
We know nothing about Fen'Harel's ability to absorb the soul of another being, so I doubt that's what happened. Apart from that, Solas/Fen'Harel is clearly not malavolent - and while he did not say the entire truth about his nature, I don't think he would kill Mythal, especially in such a treacherous way - it would go against everything we learned about his character.
- Why was the Inquisitor at the Temple of Sacred Ashes in the first place ?
- Why the Anchor has permanently been attached to the Inquisitor ?
- What good has the Inquisitor did to defeat Corypheas, just sent him to the fade and he is undone ???
1. Depends on character's origin. It is described when you pick the race and class in a short summary.
2. A mistake. The Inquisitor touched the elven orb and something went wrong, attaching the Anchor permanently to his/her hand.
3. Inquisitor opened a rift inside Corypheus body. It's safe to say that it killed him and sucked his dead body into the Fade.