Since Eowyn has many facets to her, it would be hard to analyse every aspect of her. The aspect I want to pick on, I think, is why she dressed up as Dernhelm. A quick answer to this is: she wants to be a guy.
Honestly, I don't recall when we meet her that she is satisfied being a woman. When asked what she feared, Eowyn gave a description of it. I think that this description is her perception of the life of a woman, any woman. She thinks that women have nothing to do, stay at home, and with nothing exciting happening except when the men return from war or the hunt. Eowyn would make a good contemporary American woman and a lousy hobbitess.
Therefore, I think Eowyn wants to be a guy, or at least return to the days of Shield-maidens. She has this nagging desire, and I think that she thinks that if Eowyn could prove herself a proper warrior, then she would feel 'whole'. I imagine that when she met Aragorn, that she didn't love him or even seek a great relationship, but she loved the idea of what he represented and what he would do. He was a future king, a hero in Rohan and Gondor, has a proceding reputation, and is the guy that any smart girl would want to marry and any father-in-law would be proud of. So I think Eowyn didn't love him, but maybe was jealous of Aragorn and wanted to be a great man of honor like him.
It could be she was jealous because she felt useless, like she feared. She wanted to do something to make people proud of her, and that's a hard thing to do (in her mind) when you're women. What Faramir did was show that Eowyn was a person entire, and that she didn't need to prove herself. I don't think that she then depended on Faramir to make her whole, but rather he showed Eowyn that people loved her the way she was.
Thanks Tigerlily for this discussion.
~M