Great questions! I'll tackle your second bullet point first, as looking at this era through the lens of strength will only get you so far. To really understand various motivations, it requires not just some understanding of political philosophy within Japan, but also that you prepare yourself for some Shakespearean-level soap opera Game of Thrones drama.

Scholars within Japan have debated the political and spiritual role of the Emperor basically since before even the first Shogunate took political and military control of proto-Japan, as the Fujiwara clan were ruling as perpetual regents at that time. As a very general rule, opinions regarding political philosophy tended to favor support for the contemporaneous power structure, or the one which would replace it in times of turmoil.

During the Edo period, the Tokugawa styled themselves as caretakers of the state on behalf of the Emperor, who could then have the leisure to focus on all the important spiritual/symbolic/ceremonial things that only the Emperor could do. They were doing him a favor, you see!--or so the reasoning went.

Because of old grudges that basically got deepened at the end of the Azuchi-Momoyama period, when the Tokugawa rewarded their backers and punished their enemies as they divided up the conquered territories, the Satsuma and Choshu domains continuously hated on the bakufu over the course of those centuries. Moreover, they got away with a certain degree of insubordination because they were far enough away from the center of power that the Shogun could do little about such things unless he wanted to start a massive mobilization and pitch the country in protracted war. As a result, those domains managed to keep a fair bit of power to back up their grudge, and remained voices of critique within the system.

One such critique was voiced within the Neo-Confucian Mitogaku school, which by firmly asserting the authority of the Emperor as a Confucian-style leader for all Japan, they were subtly suggesting that the Bakufu should have less authority compared to him, as they were all meant to be serving him, right?

You'd think, therefore, that the logical answer to your question is found: Folks who hated the Shogunate could rally around the authority of the Emperor and claim the moral high ground, saying that they were seeking to remove corruption and restore the proper ethical order. This even had much historical precedent, as this was a well-used tactic throughout Japanese history for those with a decent amount of power who wished to topple the status quo and put themselves in power. In fact, this is exactly what those of Satsuma and Choshu did, claiming sympathy with the many folks who were using the slogan "Sonno Joi," which means, "Expel the Barbarians; Revere the Emperor."

That said, the situation was so much more complicated than that. Here I'll need to delve a bit into the internal politics of the Tokugawa family.

The main branch of the family was the one which supplied the shoguns, but there were additionally two other sets of Tokugawa lines: the Gosanke and the Gosankyo. Originally, the Gosanke were meant to be extra sources of potential heirs to the Shogunate in case of emergency or necessity, as they were descendants of Ieyasu's younger sons. Each of the three Gosanke families were in control of their own domain. The first Shogun to come from the Gosanke, Yoshimune, then created the Gosankyo as further dynastic insurance, but they held no domains. Their sole purpose was to create potential backup heirs to the line of succession.

Since the Gosankyo in theory had no interests other than the well-being of the clan, they were the ones from whom heirs were drawn when needed, and the Gosanke essentially were barred from the line of succession.

The internal politics of the Tokugawa are even more complicated, but this sets the stage well enough for the next bit:

The 12th Shogun Ieyoshi was in office when the Perry's Black Ships rolled in. Ieyoshi was generally considered a conservative Shogun, but politically able. His son, Iesada, on the other hand, was chronically ill. Some speculate that he had cerebral palsy, some that he was mentally handicapped. Either way, most everyone near the seat of power knew there would be trouble when Ieyoshi passed if another successor wasn't named. At the time, there was suggestion that the able Tokugawa Yoshinobu would be adopted by Ieyoshi and named his official heir--but this was rigorously opposed by a political faction led by the top advisor to the Bakufu, Abe Masahiro. The question of succession was still up in the air when Ieyoshi died--right in the middle of negotiations with Perry.

The Bakufu, already under pressure during these negotiations from American technological superiority, were further hampered by the emergency succession of someone ill-fated to lead at a time when a strong leader was desperately needed. The result was the uneven Kanagawa Treaty, finally opening up the nation to trade with any Western nation besides the Netherlands.

The tremendous shock of this outcome is generally well-known, even by non-specialists. In the resultant upheaval, the political enemies of the Bakufu saw an opportunity, using fear of outside invasion to promote anti-Tokugawa sentiment, who had already demonstrated themselves as weak to the foreigners. This is but one way that Tokugawa internal shenanigans led to their decline.

An important aside about Tokugawa Yoshinobu, who was being considered as Ieyoshi's successor: he was generally considered a man of high ability, but he was controversial within the circles of power. You see, while he was adopted into the Gosankyo house of Hitotsubashi, and thus considered legitimate for succession, he was actually the son by blood of Tokugawa Nariaki--the head of the Mito domain, and one of the Gosanke branches of the Tokugawa. In other words, Nariaki and his descendants were disqualified from being Shogun as Gosanke, but in maneuvering his seventh son into adoption by the Hitotsubashi, he finally was able to have the chance to realize a Mito man leading the nation. The fact that Yoshinobu was himself shaping up to be a young man of high potential only further increased those chances.

The domain of Mito should sound familiar, because it's the birthplace of Mitogaku: the afore-mentioned school of Neo-Confucian thought that emphasized the authority of the Emperor at the expense of the Bakufu. Mitogaku, I should note, was heavily supported by Nariaki and the government of Mito.

Why would Nariaki support a school of thought that was opposed to his family?

Because he was a wily and ambitious political genius. Nariaki, as head of the Mito, had climbed as far in life as he possibly could under the status quo. Since neither he nor his heirs could ever legitimately gain actual control of the Bakufu, one of the ways he sought to increase his family's chances was the destabilize the current regime. Thus, you had a frustrated branch of the Tokugawa family supporting the academic and social movements that were also championed by the enemies of the Tokugawa, Satsuma and Choshu.

Since the feeble Iesada was in power, all of those with an interest in further weakening the main branch of the Tokugawa were further encouraged.

Into all of this mess steps Ii Naosuke, a very talented conservative politician who managed to gain power by convincing the Bakufu to reopen a high-level political position that was exclusively for members of the Ii family. The Ii family, it's worth mentioning, was generally influential, but rarely at the highest levels of power. The post he stepped into, Tairo, was only held three times from the year 1700. So you can imagine the sort of political acumen it took just to even become a major player, much less the de facto leader of the Bakufu.

But leader he was, and he did his best navigating between the demands of various nations, Tokugawa reformist factions who despised him, and the savvy Emperor Komei, who was more than happy to subtly encourage the Sonno Joi movement, seeing it as a way to regain political power lost by his ancestors.

One of his major decisions was settling the next succession crisis: Iesada only lasted five years and had no heir. The reformists once again pushed Yoshinobu as successor. Conservatives favored Iemochi. Ii Naosuke maneuvered things such that Iemochi took the throne, a wildly unpopular result that further increased tensions and resistance against the Bakufu and the conservatives in control of it. This only increased as Ii Naosuke sought to purge his detractors from government, including placing Yoshinobu under house arrest. It wasn't long after that the Tairo was assassinated by "Imperial loyalists," and all hell broke loose.

Iemochi was not much more than a child, and Yoshinobu got himself named his "Guardian" and essentially took de facto control of the Bakufu. Now, though, he faced a dilemma: he was in charge of the very political structure he and his father spent so many years destabilizing to get Yoshinobu access to this power. He spent much of this time fighting, and when he was finally named Shogun at the death of Iemochi in 1866, he was so busy struggling that he never actually spent time in Edo during his tenure as the highest political power in the land.