I can answer some, but not all, of your questions. That's because some are specific to domestic US politics (which is not my area of expertise) while others are about Cold War international relations (which I have done some reading on).

Before answering your question on Nixon's motives, I feel it's important to give you a few qualifications, as this will answer several of your other questions.

It's important to qualify that Nixon did NOT endorse the One China policy. The wording of the 1972 Joint Communique between the USA and PRC (the Shanghai Communique ) is very careful and very precise. Both sides laid out 'their side' of each issue separately, rather than presenting a shared view of the status of China and Taiwan. The US delegation stated that the US accepted that both those on the mainland and those on Taiwan considered there to be only one China. It did not state that the USA believed there to be only China, not did it express which government it regarded as the legitimate representative of one China. Nixon left the USA's position ambiguous in the hope that this would provide leverage for future negotiations. It was Carter who formally switched US recognition from the Republic of China (Taiwan) to the People's Republic of China (mainland China) seven years later.

Second, Taiwan was not a democracy in 1972. It was a military dictatorship, governed by Chiang Kai-shek, an army general who seized power in 1927 and only held a single national election in 1947, which was boycotted by virtually all opposition groups by virtue of the fact that it was held during the middle of the Chinese Civil War. Thus, in terms of US ideals, the only difference between the governments of the two Chinas was that one was a non communist dictatorship friendly to the US and the other was a communist dictatorship that was hostile to it. This was obviously an important distinction during the Cold War, but Nixon (unlike earlier Cold War presidents) recognised that communism was not a monolithic force directed from Moscow, but a broad and hotly debated philosophical tradition with many national variations, and that individual communist states has different national interests. Hence, Nixon and his National Security Advisor, Henry Kissinger, adhered to the doctrine of realpolitik: considering the pragmatic needs of your diplomatic partners and rivals and how they serve your pragmatic needs rather than their ideological alignments and your own ideals.

Which brings us to the answer to your main question: Nixon traveled to China and agreed to the Shanghai Communique because it served his foreign policy objectives.

Nixon's overriding objective was to extricate the USA from the Vietnam War without seeming to abandon South Vietnam or to surrender. Thus, he sought to gain the support of China and the USSR in pressuring North Vietnam into negotiating peace. Kissinger called this 'linkage': connecting your offer of concessions (e.g. opening talks to recognise China at the expense of the relationship with Taiwan) with a concession by an opponent elsewhere in the world (e.g. China pressuring its ally North Vietnam to abandon its plans of national unification and to accept the separation of Vietnam).

Nixon also recognised the Sino-Soviet Split as an opportunity for the USA. The Shanghai Communique would exacerbate the split in the foreign policies of the PRC and USSR. It would highlight that the capitalist USA and Communist China were capable of peaceful relations, and thereby expose how the USSR did not necessarily control or lead the communist world. Indeed, given Chinese security concerns regarding the USSR, Nixon and Kissinger hoped they might be opening the door to a new prospective ally against the USSR. And at the end of the day, the USA's alliance with Taiwan was ultimately a product of the desire of Truman and Eisenhower to contain Soviet communist geopolitical influence, so in this regard, moves towards recognising China were a continuation of the Truman Doctrine of containment.

Main source: John Lewis Gaddis, The Cold War: A New History

I've also drawn on arguments by Jonathan Fenby, Modern China: The Fall and Rise of a Great Power

Finally, it's really worth reading the text of the Shanghai Communique itself to see the nuances of the Nixon Administration's position: http://www.taiwandocuments.org/communique01.htm