Re: How high would one have to be to see the curve of the Earth?

The difficulty with Earth is that there is topography - for the Rocky
Mountains you would have to be higher than over the ocean.  With no
atmosphere, it would be possible to see your horizon deviate from a
straight line fairly easily as long as there is nothing in the way (such as
a mountain, or an ocean wave) no matter what height you are at.

However, seeing your horizon curve doesn't give you a sense as to whether
or not you are in fact standing on a sphere.  You could be at the center of
a disk, which would also look as though your horizon curved.  

There are ways to test, of course, that this is not the case - an obvious
one would be to go to a stationary point above the earth (if you can find
one!) and watch it rotate around one complete rotation.  If you are on a
spherical planet instead of a disk, the horizon you see should remain the
same size through the whole rotation.  If you were on a disk, it would
instead decrease as an oval to a straight line, and then increase as an
oval again to a round circle.