Why do we say use foreign terms such as Caliph and Caliphate or Khan and Khanate but English terms like emperor for Emperor of China and Emperor of Japan, rather than Huangdi of Zhongguo and Tennou of Nihon?

Let me take this a bit further.

Huangdi is modern Mandarin, which may have been the language of the Qing, but then if we're talking Ming it should be more like huangdei. If we're talking Tang then ɦwɑŋtei and ɢʷˤaŋtˤek for Zhou. For the Qing it's actually more complicated because if we are talking Early Qing then maybe the Manchu dərgi ədʒən is more appropriate.

Obviously this gets a little absurd, but I think it's a relevant point: Right now you see some arbitrariness in the way the titles are applied. But if you refer to all past Emperors of various dynasties as huangdi, you're just refocusing the arbitrariness without actually getting rid of it.

In reality, "Emperor" is a pretty appropriate title for China. It isn't for Caliph. The notion of Caliph has a lot of stuff tied to it that you aren't going to even get close to with a term like King or Emperor. It signifies a specific line of succession from the Prophet Muhammad. It's significant as a religious and political leader, in a way that Emperor isn't. Even with whatever religiousity might be assigned to an Emperor in, e.g., Japan, the referent is distinct enough as to not warrant a common term.

As to why not Zhongguo, the short answer is because Zhongguo doens't mean what "China" means in the many different contexts "China" gets used. I've written on this point a fair bit here, this being the first to come up after a quick search.

The tl;dr is that zhongguo as a name of the political entity is quote new. Zhongguo is an abbreviation of a couple much longer names, and in the earlier uses of "zhongguo it didn't refer to the country. It referred to the central region of the area.

Additionally, we should not forget the fact that language isn't logical, but there are potentially simple answers. Redundancy is built into language, so there's a certain amount of noise that can be tolerated without losing the signal. If I said the King of Iran, and if that's what we called him, then great you'd understand and it'd be fine. But there was perceived (though not necessarily consciously so) value in having further disambiguation by calling him the Shah. Were it cumbersome to do so, it wouldn't have stuck.

Also, don't underestimate the habit of writers (media or otherwise) to take full advantage of how foreign titles make the scene all that more exotic. If we just said the President* of Iran, that's conveying a very different feeling than calling him the Shah. That's a bit outdated now, or at least I would assume so, but the way the East was viewed by the West (as much as those terms can be applied) was nothing if not as the Other.

edit: * I don't mean that the president is the Shah. I'm saying there is an inherently exotic (Orientalist) sense to titles like Shah for the average English speaker as compared to things like "King" don't have and that hypothetically were we to call him that there's a sense of exoticness to it. I specifically said "president" to make the point that titles of foreign rulers don't need to actually match the local historical usages. I know Hassan Rouhani isn't the Shah.