Re: My daughter doesn't fit in
There's a book called "Quiet: The power of Introverts in a World that can't stop talking". Have her read it, you and your husband read it also.
First thing she needs to learn is she has incredible value to the people she loves and the world in general. Just by how you described her I would bet her emotional depth is much greater than most people, and that's one of the things that make it hard for her to connect. Most people live their life superficially, nobody wants to always be thinking and feeling, yet introverts can't help it, she needs to understand that's OK. She may go thru her life with only a handful of friends, but the depths of those friendships are what's important.
I very much relate to your daughter, even though I'm now a grown man. Both my kids went thru exactly what your daughter is now going thru. The hardest thing to do is be the first to reach out, you mentioned these people never invite her to do anything, I would bet she never invites them to do anything either. Seeing how my kids were I forced myself to initiate social situations for them, heck when they were young I would call the other kids parents myself. Have a bon fire, take the kids to an amusement park or swim park, treat them to a concert, take them camping or rent a cabin for the weekend.
Another thing I did with my kids was hire a life coach, that was a great experience for them. She would give them homework forcing them out of their comfort zone. Some of it was learning it is Ok to be alone, go to a movie or go eat at a restaurant by themselves. Other times she would pretty much force them to initiate contact with someone, while in her office she would have the kids call someone and make plans.
Maybe I'm off the mark with why you daughter is how she is, but you description of her fits me and my kids perfectly. Get the Quiet book, first she needs to understand there's nothing "wrong" with her, she just needs to learn to live within her parameters.