Katmandu to Lhasa overland - then into Mongolia

I want to travel from katmandu to Lhasa overland and end in Mongolia. I understand that a group visa is required, but that the group member all must exit at the same place. Therefore, is it currently possible for me to be on a group visa by myself so that I can leave by myself? (I will be traveling by myself) How long do the group visas give you within the country?
What is an average price (per person) for an overland tour from Katmandu to Lhasa? I just need a ballpark range, say, for a general 5-7 day tour ending in Lhasa. I assume that a travel agent can pair me with other going that way, how long does one usually have to wait for enough people, say, in mid-September? Is it possible to arrange the group tour if I am to be on my own personal group visa?
How much is the train from Lhasa to Beijing?
Can you recomend some reliable travel agents that can arrange overland trips in Katmandu, so that I can contact them and start working out details before I arrive?
Lastly, and perhaps most importantly, for those who have done this journey, did you enjoy it? I am sure the scenery is spectacular. But did it feel too touristy? I felt that much of what I've seen of China in the east was quite contrived and touristy, but I suspect that much of Tibet is [hopefully] more interesting and authentic. Or have the Chinese destroyed it? I expect Lhasa to feel a bit like a tourist extravaganza, but I am hoping the that more remote areas on the way feel more genuine. what do you think?
Thanks,
Hal fischer

Once you leave Tibet, generally you have 7-10 days in China to depart the country. There is at least one report of someone getting another 7-10 days in an extension but I would not count on this.

Most people do NOT recommend going in this direction because of the visa uncertainties. See Losang's Tibet FAQs.

For recommendations in Kathmandu, try the Indian Subcontinent branch which covers Nepal.

Ruth

Thanks.
Hal