Just for the record, my 360 is sitting vertically and have no real heat problems. It does get a bit toasty, but who's doesn't? As for why the bottom is hot, not sure. Heat rises, so I don't see any reason why the bottom would get warm. I haven't lifted mine to check the heat at the bottom, but I wouldn't have thought it'd be an issue.
I haven't opened a unit yet or seen the layout, so I don't know what's at the bottom of the case. I'm sure someone here could tell you.
As for the best way to sit it, as some have said, it'd be more stable horizontal but it depends on the space you have. What's best is down to what works for you. The 360 was designed for either way so it should work either way.
Just find somewhere with plenty of ventilation and go nuts.
Enjoy 360 gaming.