Do tectonic plates "float" over the mantle and "collide" like icebergs?
you are missing a big factor, the plates are not moving due to the momentum of an initial impulse. They are being actively moved by the push and pull of mantle convection. Much like how icebergs are pulled along by ocean currents. the iceberg analog however breaks down because icebergs melt before they can do much complex interaction, where as continental plate material is more or less permanent. also icebergs are subjected to far lower forces than continents, a closer analogy would be ice in glacier. 

The amount of energy the mantle is supplying is massive, far far more proportionally than an iceberg is subjected to, more than enough to drive continental plate together and build mountains.  if you are interested in the detailed mathematics you can start here. 

The continents float on the mantle becasue they are less dense, but they are moved by mantle movement transferring energy.