Fantastic question. I can only answer the China-related portion of your question.

In short: Not really, but the government expected it to be unpopular and acted accordingly. Nixon's decision to begin talks with the PRC (and Carter's decision to diplomatically recognize the PRC, not the Republic of China as the sole government of China) in hindsight looks very daring, but public and Congressional reactions were rather positive. I'll give a few telling examples:

Let's first look at public opinion polling from the time (source). While extensive research (Isaacs, 1958; Cohen, 1978; Kusnitz, 1984; Mackerras, 1989; Mosher, 1990) has shown that U.S. opinion toward China closely tracks positive and negative developments in the relationship and this is another example. A Gallup poll from September 1954 (just after the Korean war where the U.S. and China armies fought each other), shows that 13% of Americans had favorable views of the PRC, 74% unfavorable. Jumping ahead to 1967 (only 2 or 3 years before Nixon and co. began back channeling with the PRC), polls showed 5% favorable/91% unfavorable, but the year after Nixon visited China polls showed that opinions had drastically changed with 49% of Americans favorable, 43% unfavorable. Clearly, U.S. public opinion, at least on this, is highly dependent on elite signaling.

To show how apprehensive Nixon and Co. were at the time, look no further than the charades that the U.S. government went through to send National Security Advisor Kissinger to Beijing in 1971 for his now-famous secret talks with PRC Premier Zhou Enlai (source and source). Starting in 1970, Kissinger began back channeling with the Chinese via the Pakistani's. For the next year and a half, the two governments continued this secret method of communication for fear that it would leak to the public that their government was dealing with an adversarial state. Finally, in July 1971 Kissinger, along with a small entourage of staffers, began what was ostensibly a public trip to South and Central Asia with the final leg being in Pakistan. Once in Pakistan, Kissinger faked a stomach ache, left a state dinner, and was driven by a senior Pakistani military official in the dead of night to a military airport. There, he and his entourage boarded the Pakistani president's personal plane and flew to China. The story has other layers including recruiting Pakistani officials to fake calls to Kissinger's empty hotel room and interviewing several Pakistani doctors who would corroborate the fake stomach ache. These, in my opinion, are not the actions of a government that is confident in the public's support for its actions.

Intriguing story aside, you only need to look at what Nixon and other important U.S. officials have said (either at the time or in hindsight) to see how cautious they were of public opinion and then how surprised they were at the public's positive reactions.

Nixon, speaking with Kissinger and Zhou about Taiwan in February 1972 stated:

"One thing that is very important – and I know Prime Minister [Zhou] with his understanding of our press and Congress will realize this – I must be able to go back to Washington and say that no secret deals have been made between the Prime Minister and myself on Taiwan. So what I must do is to have what we would call ‘running room’ which the communique language I hope will provide, which will not make Taiwan a big issue in the next two or three months and the next two, three, or four year so I can continue to do the things to move us toward achieving our goal [of normalization]…

I would simply close by saying that I can [reduce U.S. military presence in Taiwan and normalize relations with the PRC] without question in my mind…if I can do it gradually but inevitably. But if I were to announce it now, it would make it very difficult to do, because it would raise the issue at the wrong time.” (source)

Winston Lord, then an advisor and speech writer for Kissinger and later an ambassador to China, in a wonderful oral history (source) stated:

"I don't recall any tremendous pressures from the right wing, or China lobby. I'm sure that there was some concern expressed to Nixon and Kissinger. Certainly, the overwhelming reaction from the media was positive, as was the overwhelming reaction from everywhere. American public opinion and Congress were an easier sell politically than we thought would be the case. I'm sure that Nixon, in particular, was somewhat nervous about public reactions as we went along including his conservative base. This initiative did take some courage. I'm sure that Nixon was pleasantly surprised at how limited the negative reaction was. At least, that's my clear recollection."

By 1972, Nixon and co. had done a 180 on their thinking. Seeing the positive public reactions to his dealings with the PRC, he embraced it and used it in his re-election campaign. The 1972 RNC party platform states that before Nixon:

"The Western will was dividing and ebbing. The isolation of the People's Republic of China with one-fourth of the world's population, went endlessly on...Now, four years later, a new leadership with new policies and new programs has restored reason and order and hope...We have moved far toward peace: withdrawal of our fighting men from Vietnam, constructive new relationships with the Soviet Union and the People's Republic of China, the nuclear arms race checked, the Mid-East crisis dampened, our alliances revitalized." (source)

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I hope this answers part of your question. Feel free to message me or comment with any follow ups.

This is a big question, but I'm going to try to tackle some of the angles of the popularity of "Beginning the process of winding down the Vietnam War." Primarily working off of Rick Perlstein's book Nixonland, supplementing other sources throughout.

The first thing to make clear is that Nixon's public position was never in favor of perpetual war in Vietnam. By the start of 1968 the peace movement was a large force in American politics, one that did not obey party lines. Nixon knew this and did not adopt the attitude of a total hawk. During his time as a newspaper columnist he attacked President Johnson for the perception that he was proposing a unilateral drawdown of troops, weakening the hand of the South Vietnamese in favor of the communists. But by early 1968 the war was becoming distinctly unpopular, with opinion polling showing a plurality of Americans saying sending US troops in the first place was a “mistake” shorty after the Tet Offensive, a number that would continue to grow as the election and Nixon’s eventual presidency went on. So when Nixon ran for president, he ran on the line of "Peace with Honor,” wrapping up the war on favorable terms, but promising that he would bring US combat operations to an end. While pulling out of the war and leaving SV to collapse was unpopular, continuing the war on its current trajectory was also unpopular, especially given the knock-on inflationary economic effects the war was beginning to have.

The second thing to make clear is that Nixon did not begin the process of winding down the Vietnam War. Lyndon Johnson began that process in 1968, sometime after he resigned from the presidential race. Negotiations were underway between the North and South Vietnamese in the fall, with Johnson hoping to announce a major breakthrough ahead of the presidential election and save the Democratic Party’s control of the government. Publicly, Nixon was in favor of these negotiations, to the point of using his support of the president’s negotiations to dodge questions about what his Vietnam strategy would be. One of several quotes by Nixon on this matter: "If there is a chance we can get the war over before this election, it is much more important than anything I might wish to say to get you to vote for me...I will not make any statement that might pull the rug out for under [President Johnson] and might destroy the possibility to bring the war to a conclusion" (Perlstein 350-351, quoting Nixon’s comments to the American Association of Editorial Cartoonists). Privately however, he was working to sabotage these peace negotiations, as the private notes of his deputy Bob Haldeman confirmed about a decade ago. The Nixon campaign worked through back-channels to tell the South Vietnamese negotiators that the Nixon administration would get them a better deal than the Johnson administration could, leading the South Vietnamese to boycott the peace talks. Given the incredibly slim margins of this election and the growing unpopularity of the war, this sabotage likely cost Hubert Humphrey the election.

After he became president, he sought to achieve his Peace with Honor by escalating the war. The strategy was to degrade the capacity and will of the North and the Vietcong to fight so they would agree to better peace terms. To this end, Nixon drew up plans for operation Duck Hook, a new assault on North Vietnam that would be carried out if progress was not made in peace negotiations by November 1969. This plan contemplated, among other things, bombing the dikes to destroy the country’s food supply and the possible deployment of nuclear weapons (Seymour Hersh, The Price of Power 124, William Burr and Jeffrey Kimball, “Nixon’s Nuclear Ploy”). Nixon ultimately pulled back from this course because of the declining popularity of the war, writing in his memoirs, "Although publicly I continued to ignore the raging antiwar controversy, I had to face the fact that it had probably destroyed the credibility of my ultimatum to Hanoi." It was this unpopularity of military escalation that made him feel the need to pursue the bombing of Laos and Cambodia in secret.

In terms of backlash, Nixon did not face public protest for his de-escalatory moves - drawing down troops, negotiating with Hanoi, turning more of the fighting of the war over the South Vietnamese military - whereas the country was facing significant non-violent and violent protests as backlash against the continuation and escalation of military operations. Massive non-violent protests were organized in the fall of 1969 by the Vietnam Moratorium Committee, which helped scuttle the threat of Duck Hook, and the riots that led to the Kent State Massacre were part of a series of such campus riots taking place against the then-revealed bombing of Cambodia. While Nixon identified his base in the Silent Majority in support of conservative policies, it was not a base in favor of perpetual US military operation in Vietnam, simply one opposed to pulling out in defeat. This is why the policy Nixon ultimately pursued, Vietnamization, was one that would support the war effort while reducing US involvement and the social ferment that involvement had brought domestically.

In short, Nixon came into office significantly running on the promise to end the war, but to end it on favorable American terms. To do this he sabotaged the negotiations of the sitting president because he believed that winding down the war would make the president and his party popular enough to win the 1968 presidential election. Throughout his first term, Nixon was significantly fighting against popular opinion in his moves to continue the war, receiving popular support when he appeared to be drawing US involvement in the war to a close.