http://www.crazyguyonabike.com/ has a section on traveling with pets, personally I like this blog:
http://cyclinggypsies.wordpress.com/
i think any bike will do, its just a matter of personal preference (comfort vs weight etc), but hauling all that extra gear for the dog is going to require a trailer (or perhaps a Big Dummy?)
you might want to check up on the visas too, china and mongolia could be quite tricky especially with a dog, on top of that for Lhasa you need an extra permit for Tibet, I dont even know if you can get one as a cyclist, I only heard stories from people who did without (illegally) but apparently checkpoints have tightened (and a dog will surely catch some attention
)
best of luck!