What would Tibetan society and government look like? as the leader of an actual sovereign nation what would Tenzin Gyatso be like? All in all what would an independent Tibet be like?
First off, Tibet would be a theocracy. In the early 20th Century until (I think) the 1940s, Tibet was still keeping serfs in a state of slavery and would amputate limbs as punishment for offences. The 13th Dalai Lama attempted to bring Tibet out of the Dark Ages by instituting a secular education system, along with the religious system. The penal system was revised and made uniform throughout the country. Capital punishment was completely abolished and corporal punishment was reduced. Living conditions in jails were also improved, and officials were designated to see that these conditions and rules were maintained. The Dalai Lama also expressed a interest in bringing automobiles, electricity and the telephone to Tibet and in 1931, a hydroelectric system was even installed in Tibet. If the PRC or the ROC left Tibet alone, I think these reforms would continue, but I think the Dalai Lama would still maintain his role as a absolute monarch within the Tibetan government. The Tibetan government only became democratic after it was kicked out of it's own country. (See the Central Tibetan Administration).
Second, no one in the world recognized Tibet as a independent nation. This was because, for all intents and purposes, Tibet was a renegade Chinese province. In 1912, the 13th Dalai Lama was even promised a place within the Chinese government, but flat out refused. The only reason why Tibet was allowed to exist for 38 years as a independent country was because the rest of China was tearing itself apart in the '20s, 30s and 40s'.
But, this discussion is academic, since the PRC would never leave Tibet alone. Why? Tibet's mineral wealth of $130 billion (made up of iron, gold, lead, lithium, mercury, uranium, etc) would be far too attractive to ignore and also, because the Yangtze and Yellow Rivers start in Tibet (These are the most important rivers in China). Also, the Himalayas provide a natural defence against invaders from India. That goes for the ROC too, but at least with the ROC taking over Tibet, Tibet's cultural heritage would be respected and you won't be seeing crimes against humanity being carried out against Tibetans.
If you want a independent Tibet, then I'm afraid post-1912 is too late a POD. If you want a independent Tibet, then Britain has to declare a protectorate over Tibet during the Younghusband expedition in the early 1900s.