So, bullet points because I need to study:

I think we need to pull this discussion as far back as the seventeenth century and the Kingdom of Sikkim.

Sikkim was taken over by the British in the early 1800s for various reasons (to take tea-fertile land of the Himalayan foothills, to open "the high road to China," to build the empire, etc.) and the way the treaty was written left Sikkim open to be controlled and annexed by India in the future.

This happened circa 1975 when chaos opened the door for India to invade and allow Sikkim (long story short) to vote for annexation. Sikkim did so, and as required by India's constitution, all states are required to have a republican form of government. Thus officially ending the rule of the Chogyal (Dharma King).

Since previous to that event Sikkim was just a "dependency" of India, and previous to that was a tributary of Tibet, China laid claim to the region and Chinese soldiers were claiming that they would march to Sikkim and liberate it in the same way they liberated Tibet in the '50s after they marched into Tibet.

It wasn't until 2003 that China fully accepted India's claim to Sikkim, though, as you can see, don't fully agree to the exact border.

Bonus questions that may be a little out of the purview of historical analysis: how does India/Bhutan/China view the history individually?

This is pretty much the question. The Doklam region wasn't really on any one's mind until, well, recently. Historically the area is basically just a barren border region home to traveling merchants and nomads going from Bhutan and Tibet to India and Sikkim. The Bhutanese and Tibetan governments have historically had an elaborate system of laws and property rights just like India and China, but as far as I can tell, very seldom demarcated strict boundaries and lines until a conflict arose that demanded one.

This became a big issue in the summer of 1914 during the Simla Conference. Representatives from the Republic of China, Tibet (run by the 13th Dalai Lama), and Britain met in Simla in the Himalayas to figure out the relationship between the three of them and the exact borders. It was George McMahon who drew what we now know as the "McMahon Line" which (basically) delineated the southern border of Tibet, obviously in favor of Britain.

The Chinese (who came woefully unprepared for the conference, the Tibetans came with volumes of books featuring tax receipts to lay claim to territories they regarded as belonging to Lhasa) ended up leaving the conference angrily without signing the agreement. The treaty was signed by Tibet and Britain, with a note stapled to the front saying that the Chinese walked out.

Here's where today's problem truly begins: the People's Republic of China basically inherits all of the borders and agreements of its predecessor, the Republic of China. In the same way, the Republic of India inherits all of the treaties, borders, and agreements of its predecessor, the (British) Government of India. If Tibet ended up independent, no problem. Tibet and India had already agreed to their border. Buuut the Chinese walked out of Simla and laid claim to their original borders, which were farther southwest and less defined than the McMahon Line. This meant that, well, since they never signed it, that the Chinese had no intention of following through with the agreement which they did not agree to. And when the Republic escaped to Taiwan and the People's Republic came around, claiming all of the Republic's claims were now the People's Republic's (I.e. the regions that were defined by British geographers as now Indian).

tl;dr, Tibet and Britain agreed to a border in 1912, welcomed the Chinese to also agree to it. China disagreed. Fifty years later China took over Tibet and did not hold itself to the agreement that the Tibetan government had signed in 1912.

Can historical analysis help us make a claim for the region today?

Not really. The area is mostly a highway used by nomads. It's either Tibetan or Bhutanese, but this has more to do with modern geopolitics and nationalism than historical ownership, which was never really relevant until the past century.

One last sub-question: where has Bhutan stood historically (i.e. the 1960's) and recently with respect to India's support - was it welcomed or reluctantly accepted?

Bhutan has welcomed Indian support though recently (Doklam Plateau aside) have been somewhat standoffish because they don't want to be reduced to the status of a client state. They call their alliance with India the "Road to Friendship." Indian military have been stationed in Bhutan since at least 1962 when Bhutanese airfields were used in the Sino-Indian War. More information here.

Hope that helps.