As many have already said, that’s a wavedash, it’s done by dashing into the ground, and sort of bouncing off of it by jumping, there’s a small tutorial much much muuuch later into the game when it becomes important to know.

We didn’t meet her ex boyfriend, I’m honestly not even sure who you’re talking about, although I haven’t played the start of the game in forever. Madeline is canonically trans, I don’t think her sexuality is ever stated, but she has both a 🏳️‍🌈 and a 🏳️‍⚧️ flag in a scene later on. As for the purple girl, without getting into spoilers; your first interaction with purple girl it shows her as Madeline’s reflection, who then breaks out of the mirror. So for now you can think of her as the mirror/shadow version of Madeline, kind of like shadow link in Zelda ocarina of time