What sorts of essays do school-age children (and test-takers, generally) in China have to write?
Only Mainland.
Nowadays, children won't write in formation.Letter? That is not the case I know.

But in history, we do. 

骈文
Sentences must be paired. Characters must be carefully chose.
After two big man of letters claimed that they shouln't write 骈文 anymore, cause it is only highfalutin, 骈文 was dead.

This is the most famous 骈文 quote from &lt;&lt;滕王阁序>>, almost all educated Chinese
can recite it.


  落霞與孤鶩齊飛，秋水共長天一色。


If we take close look at the sentence, these word are pairing. 

落霞(sunset glow) vs 秋水(water in autumn)

與: (with, simplified is 与) vs 共(and)

孤鶩(single wild duck, simplified is 鹜) vs 長天(wide sky, simplified is 长)

齊飛(fly at the same time, simplified is 齐飞) vs 一色 (one color, same color)

I can imagine a picture, in autumn, a wild duck fly above a lake or river, its background is sunset glow, the sky has same color with water surface, that's beautiful.

八股文
八股文 is a strict article. It must consist of 8 parts, 

破题: two sentences, interprets the topic.

承题: 4 or 5 sentences, talk about the key of 破题.

起讲: multiple sentences, descript the overall topic.

起股: 4,5 or 8, 9 sentences, start discussing.

中股: pairing sentences, the essential part of the whole article

后股: pairing sentence, mention what you have not talkled about

束股: 2 lines, 3,4 or 4,5 per line, pairing, restate your views, make conclusion

大结: multiple sentences, finale

The structure of 八股文 is very good at argumentative writing, later people found it limits your thinking, cause 八股文 also limits the topic which is from confucianism book. In hundreds years, all the topics you can find from &lt;&lt;论语>> and &lt;&lt;孟子>> had been used, they begin to mess things up, grab one sentence from 论语, grab one from 孟子, combinate to one, ask students to write about it.

八股文 is deprecated after the Qing dynasty.

That is the status of two main structured article in China.  Other structured texts: 格律诗, 对联.