It is only the job of the executive because Congress told them so via Acts of Congress. Looking at e.g. the firings of the inspector generals, Congress has put very clear language into its laws on why and when those inspector generals can be removed by their post, yet Trump and his cronies ignored this.

It should not be a formality because while it is true that the Republicans have a slight majority in Congress, the founding fathers never intended this most powerful of the three branches to be run by parties. The power in Congress is split up geographically for this very reason, but the party system, that secured its seats with gerrymandering, is highly toxic for a functioning legislative power in the US. It is disappointing to see Republicans in Congress not restricting the executive orders of the new self-proclaimed King.

The president can fire any executive branch officer at any time for any reason regardless of what any statute says about it.

There are two precedents for this to my knowledge, though there may be more:

  - the failed impeachment of
    Andrew Johnson established
    that the president can fire
    principal officers without
    the Senate's advice and
    consent

  - Spicer vs. Biden, which
    established that the president
    can fire officers with fixed
    terms

He was clearly trolling. Grammatically that tweet does not parse like himself calling himself a king. For all you know he loves the British king, or some other king, or maybe he was referring to Jesus. But he got what he wanted from that quip: it got reported, along with credit for ending congestion pricing in Manhattan. Why the media still falls for that, I don't know.