I'm back from traveling! So while I originally would have started this post in the 19th century, I'm going to wind it back to give a basic overview of Tibetan political history, answering both u/enlightenedark 's questions and u/glymao 's from below:

Short overview:

Tibetan history begins in the mid-7th Century with the rise of Tri Songtsen Gampo who commissioned the creation of Tibetan script, giving us the first Tibetan histories in the process. Songtsen Gampo descends from a long line of pre-historic Tibetan kings, but is considered Tibet's first emperor and under his reign, Tibetan government was centralized and the realm expanded in all directions, notably taking Nepal as a vassal state, conquering the Zhangzhung kingdom to the west, and the 'Azha confederacy to the east. More on those below.

The Tibetan Empire reached its zenith under Songtsen Gampo's descendant Tri Song Detsuen who expanded Tibet even further, conquering Chinese territory, seating his own puppet Emperor, seizing the Silk Route, and turning the country wholeheartedly towards Buddhism for the first time. About a century after him, however, the empire collapsed as a quick succession of imperial assasinations released the power of the warring clans that had been kept in check by the Emperors and Tibet entered its traditional "dark age."

Tibet entered a century and a half of very little systemic control, and then in the 1000s, Buddhism began to filter back into the country and the rise of new monastic schools opened the way for more centralized control, albeit of a theocratic variety. The Tangut (called by the Tibetans the Minyak, and by the Chinese the Xixia) who lived in modern Qinghai Province (Amdo to the Tibetans) were fond of Tibetan Buddhism and employed many Tibetan Buddhist monks for both practical (medicine and historic record keeping) and mystical (ritual empowerments and tantric rites) duties. The Mongols, who liked taking the best of cultures and systems and incorporating them into their own empire, took note of this and after defeating the Tangut, took an interest in Tibet. Long story short, the Mongols made a show of force, contacted the well-known and respected Sakya Pandita, and told him to tell the other Tibetan rulers to surrender or else. Sakya Pandita, having seen the power of the Mongols, wrote a letter to all the other rulers of Tibet and confirmed their worst fears. In this way, Tibet came under Mongol rule with very little bloodshed.

Sakya Pandita (as his name implies) was the ruler of the Sakya sect of Buddhism, and after becoming the spiritual advisor to the Khaan, was gifted all of Tibet to rule under the umbrella of Mongol overlordship. Sakya rule persisted until the 1350s when an officer of the Sakya, Jangchub Gyaltsen (Tib: 'byang chub rgyal mtshan) rose up in revolt and deposed the Sakya. Very important for historical legitimacy: Gyaltsen, who began the Phagmodrupa dynasty, could only do this if he knew he could defeat both the Sakya (which he obviously could) and the Mongols (he obviously could not). However, the Mongols were a bit busy with the Ming revolt, so when they received a letter from Gyaltsen saying he had deposed the Sakya and now requested vassalship of the Mongols, they consented, granting him the title "Tai Situ" and were then defeated by the Ming and retreated north to Mongolia.

The Phagmodrupa lineage was eventually deposed by the Rinpungpa, who never requested confirmation from the Ming and which was never offered. The Tsangpa dynasty deposed the Rinpungpa in the 17th Century and (again) were not offered and did not ask for Ming legitimacy. Finally, when the 5th Dalai Lama was placed on the throne of Lhasa by his patron, Gushri Khaan of the Qoshot Mongols, history repeated itself: Gushri Khaan was declared "King of Tibet" by the Dalai Lama, while the Khaan simply gave over the entire country to his spiritual advisor, reserving only a small part of land for himself and his tribe on the Kokonor (modern Lake Qinghai). More on all of this in a second, as above.

Politics in Tibet was fraught with violence based on systemic troubles between the Tibetans and Mongols, and eventually the Manchu Qing Dynasty stepped in to pacify the situation bit by bit. The story gets long and detailed here, but basically, the Qing knew they couldn't just storm through Tibet and conquer it. The Tibetans would resist, and more importantly, they would serve as a uniting factor for the Qoshot and the rest of the rival Mongols. Meanwhile, they still had Chinese rebellions in the east to still deal with. Until the first half of the 1700s, the Qing considered the western Mongol tribes a significant threat to their power, and befriending the Tibetan religious establishment was their best bet for security on that border. Despite all of that, we can say that by 1720, the Tibetan political situation was propped up by the Qing, and by 1792 at the latest, Tibet was militarily occupied under their own request under threat of Nepali invasion (and by "threat" I mean, they were actually being invaded at time).

Qing rule proved more difficult overtime. The reasons aren't particularly important at this point in time. Suffice to say that the Qing were quite liberal with their application of capital punishment. So much so that when the Tibetans overthrew the Qing they outlawed it.

Anyway, in 1912, both the Chinese and the Tibetans started rebellions to overthrow the Qing. Both were successful. I'm not an expert in the Xinhai Rebellion, only that it resulted in the Republic of China. In Tibet, however, it resulted in a declaration of Independent from the 13th Dalai Lama (as seen below) and the beginning of an independent state of Tibet.

Cont'd.