For the most part, Darron, they do, but people have been playing the KOTOR games too long and think that they can do other things, too. Mostly the differences in the crystal change shape, size and energy-level of the blade (most of the latter making the blade quite unstable). In truth, half the reason we have so many different crystal types is because you have varying levels of stability and focusing efficiency, not to mention ability to withstand damage (some crystals, such as the Corusca Gem could technically survive the destruction of the lightsaber housing it, for example).
And no, they can't, Darren. Lightsaber Crystals are usually imbued with Force energy when initially added to a lightsaber as part of the rituals used by a Jedi or Sith in the creation of the weapon (though a Synthetic Crystal has no need of those rituals, given that it is made using concentrations of Force energy and a few certain minerals, with a Forge), but draining that Force energy from a crystal would probably shatter it and make it utterly useless.