Sauron doesn't need to be uninvolved in the catch-and-release program for Morgoth to be the one to place the spell on Annael. It can be the result of a collaboration between them. If I remember correctly, part of the reason that Sauron tries to create his own spell is that he is tired of Morgoth taking over and, in Sauron's eyes, ruining the projects he starts. However, if we want to show Fingolfin wounding Morgoth as something that really causes a rift in their relationship, it might be worthwhile to show them working together earlier in the season.
I am not suggesting that we merely do a repeat of Edhellos's story with Annael. The reason I brought up Edhellos as a comparison was that, under Morgoth's spell rather than Saurons, I thought Annael's treachery would be more likely to take the form of direct sabotage, like how Edhellos killed the messenger, rather than gathering information.
The original plan for the four named character prisoners was to show four different outcomes: remaining a prisoner in Angband (Diriel), legitimately escaping (Rhogrin), broken and driven mad by the SoBD (Edhellos), and under the SoBD but does not know it (Annael). However, the way Edhellos's story ended up playing out, she was more in the category of being under the SoBD and not knowing it. Therefore, I don't think we necessarily need to stick to our original plan for Annael.
No. No it doesn't. The gender angle only comes up if the reason Annael is able to resist the spell and survive is because he is a male, which won't be the case. I don't see any reason why the self-exile situation you describe below wouldn't work for Annael, just with Morgoth in charge of the spell rather than Sauron.