"The kindergarten teacher sits down on the chimney and sings."
Translation:Az óvónő felül a kéményre és énekel.
So, my answer used "leül" (sits down) -- this is accepted as correct, but the other correct solution uses "felül" (sits up).
So, I get the idea of someone sitting up onto the top of a tall chimney. That makes sense to me ... but do both versions of this sentence (sit down, vs sit up) still mean, basically, the same thing in Hungarian?
Yes, basically it is the same thing, it is just a directional indicator, relative to a place that one would normally sit. I sit "down" on the grass, the curb, the bed. I sit "up" on the table, a horse, a train, an airplane, a chimney.
I mixed in a little bit of "mounting", as well. When you mount your bike, your horse, etc., you also use "felülni".
"Felülni" can also be what English means by sitting up, from a lying position.
is she singing while she is in the process of sitting down, or after she is already sitting?
Based on the order of the sentence, I would say she sits down first and then starts singing. These flying kindergarten teachers really make me think of Mary Poppins for some reason...