If cost is a primary concern, the kids can go to local school (just like Chinese kids who come to the US). The expat continuum goes from 'bubble' to 'going native'. Bubble is most expensive. Might as well stay home, no?

They can? Even local chinese kids without 户口 can't go to local schools, to hear them tell it. That was one of the primary reasons students came to the school where I worked in 2010-11.

No first hand experience, but from what I can tell, children of legitimate Z visa holders can send their kids to local schools for a reasonable fee. See shanghaiexpat.

It is also legal for expats to home school their kids (not legal for the locals).

There is a strong sense of entitlement present in American culture which leads to unrealistic expectations when living abroad. This manifests as the so-called 'ugly American'. China is a place where 'zero-tolerance' is literally a foreign concept, and anything is possible with persistence (and good connections). Which also implies the downside -- nothing is permanent or guaranteed, either.

Depends on the city. Shanghai is much more reasonable than say Beijing; each city in the country has its own often conflicting rules.

You'll have to pay some bribes but its possible. Generally, expats with kids have packages that include private schooling. Expats without packages usually have to leave when they get kids, though that's probably for the better with this air (the better private schools have huge bubbles, by the way....)

Nah, the last job I had there would have paid for schooling. It wouldn't pay for a hermetically sealed bubble of fresh air and water (and clean food), though.