Was her protege in love with her?

As to whether or not her protégé was in love with her, I'd have said no, personally, but it hardly matters in my opinion. I think the point of the film -- or one of the points of the film, at least -- is to demonstrate that we need not be so entirely reliant on our desires being affirmed (as I mentioned in another post, see the conspicuous parallels in Nathalie's subsequent recitation of Julie, ou la nouvelle Héloïse, and then reflect on all the previous disappointments she'd suffered throughout the film: the student revolution, the vulgar cover art for her series of academic texts, her husband leaving her, her mother's upheaval, and so on and so on).

So -- is he in love with her, is he not? -- I imagine that's quite well besides the point; the simple fact of the matter, as you've already pointed out, is that that love is never realized (irrespective as to whether or not it could have been) and Nathalie must overcome her disappointments in this regard. Which, by the end of the film, she certainly seems to have done! So, well done Nathalie -- here, here.

On a side note, I think you're quite right in thinking that all was not so well as it seemed with Fabien. It's been a while since I watched L'avenir, but I was under the impression that there was trouble in paradise, too: Why was Fabien on the computer late at night, and not in bed with his girlfriend upstairs, with whom he'd already established an open dialogue with regards to Nathalie's perceived feelings towards him? Not everyone in the film is shown clearly to be suffering, but I think we must remember that everyone, for a time, always is, in their own way -- Nathalie's daughter, crying idiosyncratically at the birth of her child, for example.

Why was she crying at the farmhouse? I honestly can't remember, but I imagine she had a lot on her plate then, bless her; she had a lot on her plate the whole way through, but, as I said, she overcame all this suffering in the end.

Might I leave you with a quote? I've always been dearly fond of this one from Kafka, and I think it's rather splendidly appropriate:

We are as forlorn as children lost in the woods. When you stand in front of me and look at me, what do you know of the griefs that are in me and what do I know of yours. And if I were to cast myself down before you and weep and tell you, what more would you know about me than you know about Hell when someone tells you it is hot and dreadful? For that reason alone we human beings ought to stand before one another as reverently, as reflectively, as lovingly, as we would before the entrance to Hell.

My impressions.

1. No, he wasn't romantically in love with her. The two were extremely intellectually close though, note her family teasing her about him being the son she always wanted and her husband being extremely jealous of Fabien and accusing him of using her.

I think he looked conflicted and upset driving home because they had had that argument in where he attacked her ideas and she was obviously upset about it even though she said she wasn't. He remarks on it to his girlfriend that he was unkind to her and you can tell he regrets it because he does respect and admire her. And it's quite obvious that there is no flood in her apartment, she just decided she had had enough of being there and wanted to go home so I think he feels bad that something he said made her leave early.

2. I think her crying was a combination of the two things you described. IIRC it was after they had the intellectual argument which triggered her tears. But at the same time she had had such an unbelievably rough year, losing her mother, her book deal, her husband. I don't think she expected to always agree with Fabien but he was someone she clearly adored and admired and in that one moment his contempt of her and the "loss" of the friendship (though of course it wasn't truly that) was probably one loss too many.