Jean and Llewyn

1) Jean did say nice things about Llewyn to Troy.
2) when Llewyn tells Jean that he loves her, she gives him a very sweet and sentimental look in reply.
3) Llewyn flips out when he thinks Pappi had sex with Jean (which I really doubt).

That's not in the impression, it's in the script.

I do think that in a weird way Jean and Llewyn love each other... but I'm not sure if it's a romantic love.

I thought maybe Jean, Mike, and Llewyn were all close and Jim to a lesser extent. Jean sleeping with Llewyn struck me as a way for both of them to cope with the loss of Mike.

Right when I was watching the movie Pappi commenting on sleeping with Jean made it sound like she was willing to sleep with whomever she had to in order to get her career to go somewhere. But then thinking about it and Llewyn's reaction, it sounds like she slept with Pappi to get Llewyn his gig.

There's certainly an affection there, and obviously Llewyn admits his love for her. The smile she gives him tells us it's a welcomed declaration, but we can never say if she loves him back, at least in the same way.

They have an interesting and messy relationship whatever it is. Since she's with Jim and they like to bicker with each other it seems like they're destined to stay in that gray area. Did they always fight like that? Or was there relationship uprooted after Mike's suicide? Had they slept together more than once? Were they ever an item before Jean got with Jim or Llewyn got with Diane (or someone else)? We just don't know these things, but for the sake of this story, we don't really need to I suppose. As with most of this movie you can read into it however much you want.