Is she still essential after her quest is complete?

According to http://www.uesp.net/wiki/Skyrim:Muiri yes. I'm guessing she remains essential because she's a marriage candidate.

So in order to kill her I have to do her optional quest as well to get her as a marriage candidate?

edit: never mind you don't have to do her optional quest in order to kill you just need to do the main one then her essential status is removed.

I don't think you can kill her. Not according to that page, anyway. If you're on PC, you can make her un-essential by using a mod or by using the console. But, as far as I know, if you play on console you can never kill her.

I just killed her actually but before seeing if I could kill her I saved first then put a poisoned arrow in her back and she died.