It's actually kind of nice that the up and down symbols seem to be pointing in the direction they mean. That makes it a bit easier to remember
Yes, (Though Ue is up, Shita is down)
上 - Up, Above, Over, Top, Summit, On (surface)
下 - Below, Down, Under, Bottom, Beneath
It should be. As someone else says, "on" is accepted in their constructed sentences. It seems this bit of Japanese isn't well developed.
To memorize this character well; it looks like a plant with a branch growing out of soil.