Obviously at this point the series is over and we've learned everything we ever will about the Eve character.
Did they try to get rid of the actress and she somehow managed to sue her way into another season and they didn't came up with a better solution to include her?
No, this was a played storyarc for Juliette transitioning from the fourth to fifth season:
I was really thrilled about the hexen beast story line just because Juliette had been so consistently good and thoughtful and helpful for so long.
That I thought if anyone’s going to turn and be so bad and so evil and so powerful, it should be her. So, that was really exciting for me. And then I will definitely admit that I had a little bit of mixed feelings when they – the producers called me in March and then said, “This is our idea going to Season 5” because it was hard to say goodbye to Juliette. - An Interview With Grimm's Bitsi Tulloch
That quote is dated January 2016, and so she is referring to March of 2015 of when the producers told her she would be transitioning into Eve for Season 5.
So what is her function in the narrative?
Is there any background information about the creation (and its reasons) of the Eve character?
From the same interview:
Hadrian’s Wall knew how powerful she was and kidnapped her. You know Trubel – shot her with the what we now know are tranquilizer darts and basically broke her down and turned her into like a fighting machine. So keeping that in mind, that’s one thing that the new character’s like going forward.
After her interaction with the "stick" Eve became more like the old Juliette, but it was more of removing all the emotional repression she was experiencing.
TVLINE | Eve showing signs of Juliette returning: What can you tell me about what’s really going on?
GREENWALT | Well, the feelings have come back.
KOUF | The stick opened up part of her. There are unintended consequences to the use of that stick. So, just like everything else.
GREENWALT | There’s part of her that was dead, that she had killed, that they had killed. Suddenly it was awakened. And because she’s a Hexenbiest, it had a super effect. So we’ll have to see what happens with the Eve side of her personality and now having all these feelings, which is going to include a lot of guilt. It’ll be complex. - Grimm Finale Recap Season 5 TVLINE interview
Eve was intended to be a weapon for Hadrian's Wall, and the stick ultimately broke the emotional deadening that they had done to her when they removed Juliette and created Eve.