Gotta keep this brief because I'm on my way out the door, but let me know if you have follow up questions and I can get them a bit later.

I'm under the impression that the Chinese emperor was held in power through some sort of quasi-religious concept of the world that placed China as the center of the universe and gave the emperor legitimacy through the Mandate of Heaven.

That's sort of a Western misinterpretation of what's really going on. It's not that China was the centre of the world. The term "Middle Kingdom" doesn't actually mean that but rather was meant as "central states", a part of the larger Chinese empire/s.

The Mandate of Heaven is less a matter of an omniscient Deity than it is the natural order of things. The Chinese term 命 which is translated as "mandate" also means "fate". I should be emperor not because a god is smiling on me, but because I am fated to be the emperor as I am the most legitimate contender, so people should back me.

Christianity seems to me as highly incompatible with the idea of a divinely sanctioned, non-Christian emperor who the world revolves around.

When you say divinely sanctioned, this is the issue. Think of heaven as a matter of balance and in a more naturalistic sense. This is what Taoism/Daoism gets at. The ideal emperor is one whose subjects don't even know he exists, who through basically keeping out of people's lives rules correctly. By not forcing actions, everything goes as it should.

So Divine here should really just be "consistent with the natural order of things", not actually divinely sanctioned. That this gets translated into English as "the Mandate of Heaven" isn't helping confusion among Westerners trying to understand the systems at work.

Was Chinese Christianity somehow compatible with the Confucian/imperial mindset?

Chinese Christianity has meant many different things at different times throughout the history of Christianity in China. However, people like Matteo Ricci, an early Jesuit missionary, actively positioned Christianity as not only compatible but following in the footsteps of Confucianism as a means to help sell the faith to skeptics of the time. This was in opposition to Buddhism, which he described as being a negative influence on Confucianism.

Did Christians (other than the Taiping Rebellion), who denied the Mandate of Heaven, ever pose a threat to the emperor?

Is there any reason you discard the Taiping Civil War, being that it's the most significant example of the thing you're asking about?

There were other less major cases where Christianity was the tool used by those wishing to go against the law, in that, prior to the Boxer Uprising, Christians were afforded certain special protections, and so by becoming Christian, nominally or otherwise, you were than granted this protection and had a little extra leverage in legal realms. But the most significant case of Christian doctrine being a threat was the Taiping Heavenly Kingdom. Before that, the number of Christians just wasn't significant enough to have that sort of impact.

Did emperors ever see Christians as a dangerous 5th column because of their beliefs?

If not emperors then at least other leaders. There's a reason why the Catholic Church in China isn't led by the Pope in Rome. Chinese bishops are assigned by the state, and the Vatican really has no say. It's not so much that the Christian God was dangerous to the state, but rather than the foreign Pope potentally was.