China won't let you into the Tibetan Autonomous Region except as part of an official tour party from within China. You can go around further to the north (I went that way by rail a few years ago), but most people on such a road trip would like to see India (and probably also Nepal), and to then have to go back all the way to possibly cross the border in disputed Kashmir or realistically to Uzbekistan is a bit of a detour to say the least.

(Central Asia is fascinating and I do recommend the trip through Uzbekistan and Kazakhstan at some point).

You can, but driving your own vehicle in China is extremely expensive, and it's difficult (or impossible) to enter from some countries / borders.

Most people end up shipping from India -> Thailand or Malaysia, and it's still cheaper than driving through China.

ie. see http://wikioverland.org/China the prices have come down a lot lately, but it's still very expensive.