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!
you'll probably be fine with a hybrid, altho a mountain bike would be more
comfortable. my opinion. but you might find getting spares difficult. not
a problem in the cities, but in small towns and villages, standard seems to be
26" tubes/tires....schrader. almost never see presta.
llasa can be done, but it'll be expensive. you must go in a group, with special
permits, accompanied by car and driver. it is possible to go as a group of one.
lonely planet/thorntree tibet sub-forum will have more information.
crossing borders with a dog? even if you have all the proper shot records,
the critter may have to spend weeks/months in quarantine.