No, this is an English subtly that doesnt translate well. "IN" and "ON" can be used interchangeable as colloquialisms in this sentence.
Not a very good one, that's just how we say it :)
I guess you could argue that a "farm" is mostly a 2-dimensional piece of land, and that "in" is only used for enclosed (3-d) spaces. But that argument doesn't always hold, and it's rather subjective too.
You misreading the sentence. The animals performed the act of cooking on the farm.