Yes, you only get one chance to unlock Jessica's Adultress Quirk. Quirks are like achievements, and also used to open or lock other content like it.
Chloe can only absorb Fey who are mostly energy-based (there are different kinds of Fey) and Lyx isn't one of them so no, she can't absorb her. She also can't absorb Asteria or Luna. She COULD absorb Qarinah and Roxy, theoretically, but both of them are far more powerful than she is right now.
You will be able to choose how you deal with Morgana :)
There are hints about who sent the assassin Fey, but no answers... yet ;)
I hope that answers your questions!