To your first question, yes, I think they do just 'float' around in the metal.
As for your second question, I'm pretty sure that gravity is so insignificant as to be meaningless on the atomic scale. The other forces (electromagnetic, strong nuclear, weak nuclear) are many orders of magnitude stronger. So no, the electrons will not 'fall' to the bottom of the pole. If they did, you'd get shocked every time you touched a pole ;)