Wow. Your daughter sounds a lot like mine. My daughter was diagnosed with Sensory Processing Disorder, though, but she's also very highly sensitive. She always sees the black side or the down side to life--we have to constantly keep telling her to look on the positive, that what you think about is what comes about. So if you want positive and happy and people to like you (my daughter is 13 and still keeps saying she has no friends, yet when she's around others, her "friends" are all over her, happy to see her), you must project and think happy and positive thoughts.
It is an ongoing, daily battle. We are constantly working with her to change her thoughts and viewpoint from dismal and unhappy to upbeat and happy/positive. She is shy and doesn't like to voluntarily go up to people/her friends, which could be one reason why she thinks she has no friends; her friends don't always come to her. We are working with our daughter to get her to understand that she has to put forth effort as well. That's scary for her, as she doesn't like change. Routine is more her forte.
If your daughter has a very, very hard time making friends, having changed her daycare halfway through the year might be having a profound impact on her in that the kids already have their friendships established. It's not easy for anyone to "break in" to a friendship group, and if you're shy and not very outgoing, it can be almost impossible.
I remember when I took my daughter out of her daycare when she was 5 when I started staying at home to work, I really needed to have her go to her daycare a few days/week for that I could work on some huge work projects. She absolutely refused to go back to daycare, stating that she had no friends there. When I mentioned that all the kids she'd been going to daycare where still there, she told me they already had their friends. For her, going back to daycare was not going to work.
In fact, she actually totally and completely hated daycare. She loved the teachers but didn't care much for the kids. She really, really struggled with social situations when she was younger.
I recommend the book The Highly Sensitive Child: Helping Our Children Thrive When the World Overwhelms Them by Elaine N. Aron, PhD. It's helped me enormously in dealing with my daughter. She also sees a Childhood Specialist (CS) once a month, so she can get an outsider's perspective on what she deems are "issues" as well as another take/input on how to handle them (and many times, the CS restates what her dad and I have already said).
If you can help your daughter navigate through these tricky and scary waters (as they seem to be to her), she will learn to cope with her differences. The key is getting her to learn coping mechanisms--and that's where you come in.