She doesn't quite look down on people, but she always tries to act in a way that benefits her family. Her problem is basically threefold. She is super shy and has low self-esteem, she absolutely sucks at socializing, and she feels the responsibility to act in a way that befits her family's position. Altogether this means she acts high and mighty because she thinks she should and doesn't have the confidence to step outside that comfort zone and try anything else.
Her dad sent her to a public school to try to get her to grow out of all of that, which she sort of realizes. But she doesn't feel like she should spend time with people that won't benefit her family, so she's always questioning what benefit there would be to her (and her family) if she spent time with them. She's not dissing people or even looking down on them, she's genuinely asking. She's just that bad at socializing. And when she freaks out about people being after her money or her body, she genuinely believes that based on past experience (and isn't even entirely wrong in this case). Note, she doesn't really care about wealth or anything (Fuuka isn't even remotely wealthy and they get along fine), she just judges people based on their perceived intrinsic value.
She made her request to Himeko not because she actually wants to get along with everyone, but because she's trying to make friends with Himeko but was too shy to just go up to her and ask to be friends. She wants to be friends with Himeko because she heard that Himeko is one of the two princesses so feels that of everyone at school Himeko is someone at a high enough level she has an excuse to make friends with. Midori also gets along with Fuuka for the same reason. She gets along with Routa because Himeko acknowledged him as her partner and becomes friends with him after he picks up paper for her. She doesn't really try to make friends with everyone despite her request because that's not actually her goal.
At the end of the day though, Midori is really just looking for any excuse to get along with people, which Routa sort of realizes. He tells Midori that Himeko is throwing a party with her "close friends" and that it's they're last attempt to help her make friends before giving up. Midori then gets along with everyone at the party because they're "close friends" (they're not really, but Midori doesn't know that) with Himeko and thus she can justify spending time with them and stuff.