On the school topic: Pretty much what islandofapples said about "a cry for help." Your DD's misery is a strong indication of how her current school situation is totally wrong for her. Something has to change, rather drastically.
On the coat topic: My DD was 4 or 5 when I tried to put one on her as we waited for the bus to pre-K one cold morning, and she went from happy kid to meltdown kid in record time. The only other alternative she left me was to try wrestling her into the coat in the middle of the driveway, so we went inside for a time-out instead, and she missed the bus.
Giving her a few minutes to work out her emotions was clearly not working, so I'd poke in there from time to time to see if I could say anything to defuse it, fail, and walk out again to give her some more time. Finally, I blurted out the magic words: "I don't understand why this is such a big deal. It's just a coat. You can take it off."
The meltdown immediately ended.
"Wait... you mean I can take it off when I get on the bus?"
Momentary pause while I wrestle with trying to trick her, play the boss card, or be honest, because taking it off on the bus defeats the point. I pick honesty.
"Well... I'd really rather you didn't, because it'll still be cold on the bus, when some of the kids open the windows. I'd prefer it if you waited until you were inside the school. But I can't stop you."
Problem solved. I drove her to school, and she got to explain to her teacher that the reason she was late was, "I had a meltdown."
The moral of the story: You never know what strange things might be going on in these kids' heads. Maybe your DD thinks she has to wear the coat all day, like mine did. Maybe she's afraid of losing it, she doesn't like the way it looks, or how it gets in the way of her fashion statement for the day (assuming no school uniforms). Maybe it's something else none of us would ever think of.
