I don't think it's desperation, I just think she's a bit naive in the whole 'love' dept.
It's a completely weird situation to be in, IRL no one has to fight amongst that amount of other women in such close proximity for a bloke. Especially when so many of those women have their claws out!
I feel sad for her...and him actually, as I really think they genuinely like each other but are now stuck in this whole 'game'
I don't think the faint was fake, though it wasn't actually a faint. She did start to look a bit ill and just sat down so she didn't faint. I'd probably feel the same if I was in her position.