When I worked in southern China (Nanning) as an English teacher, everyone in the office would nap at their desks or in an unused classroom after lunch. We catered to the after-school set, so we rarely had students in the middle of the day: sometimes lunch-and-nap-time stretched on for a couple of hours.

I wouldn't mind having that sort of situation now, rather than lunch and back in less than an hour.

I think my annualized salary for that year in China (2003) was something like 1/15th what I make now; but I managed to save a far greater percentage of my income. Everything is cheap in China... including life.

That is true in Japanese, but much less common in Chinese. Chinese generally has a single one-syllable pronunciation for each glyph, though grammar rules can change the tone of character in a given word. There can also be several glyphs which share a single pronunciation (homonyms): there are more characters in Chinese than there are possible phoneme combinations (given the rules of the language for constructing syllables).

There is really no relationship between latin characters and sound either, at least until you've learned them. Korean Hangul is the only character set that I know of in which a conscious effort was made to have the parts of the glyphs relate to the structure of the mouth when they are pronounced.

That being said, it is not entirely true that there is no relationship between sound and character in Chinese. Once you have learned the hundred or so base characters, these are re-used over and over as 'radicals' (parts) in the more complex characters. The main radical often gives a hint to the meaning of the character (for example, 'water' may mean that you are talking about some liquid or water-related thing) and other parts of the character often give a hint to how the characters should be pronounced.

In my experience, this is true for both Chinese and Japanese, but in very different ways. (In fact, the differences in the languages that originally shared a common writing system explains a lot of the divergence in their use of the characters.) Simplified Chinese (used for mainland Mandarin) has changed the shape of many characters without maintaining the hints that were previously embedded within the word.

My suggestion would be to learn the simple first-and-second-year hanzi for whichever dialect your children are learning... probably no more than a hundred characters or so. You can probably do so much faster than they can. But at some point the pace of their classes will increase dramatically. You may be able to keep up as they learn additional characters, but ultimately the only way to learn them is to use them: practice practice practice. It takes time.