Why is there a kara after coat instead of ni?
Because から = from. From something, from somewhere - it's always から.
I mean...that's not totally true, though.
「母にプレゼントをもらいます。」, for instance, is fine, and yet the English translation would be "I receive a present from my mother."
So why doesn't に work in this instance?
Because the flowers are physically contained within the coat. The coat isn't giving you flowers, you are taking flowers out of the coat.
Yay, you're a magician, right? to take out flower from your coat :)
What the hell is up with these weird sentences? Flowers from hats and from coats - what's next? I know it's grammar and vocab practice, but since when has that meant completely giving up on making sense?