Wow, so much to answer here...
Kids: Your kids are really young and that's probably the best age to bring. They're too young to be bored, and may well pick up the language. China can be AGONY for older kids and teens, who have to give up their social set, their cool toys, their TV, etc. and often just aren't as into this place as we are.
Medical care here frankly sucks. A dodgy, unclean, very uncomfortable spectator sport. For a family, (in fact, for anyone) I strongly advise buying expat medical insurance if you possibly can. Chinese doctors are OK for routine day-to-day stuff, but for anything more serious you should be on a plane or train to Hong Kong, Japan, Australia...anywhere but here. Or the USA, which is so expensive that expat insurance won't cover you there except for emergencies. There's a link to an expat insurance company in our Links page, and more can easily be found on Google. I DON'T have this coverage and am wishing I did...I have a 3-year-old who needs ear tubes in and adenoids out, and finding reliable care for her has been agonizing.
Bring kiddie DVDs, books, standard OTC medicines. As many as you can possibly carry.
You can hire half-day or live-in Ayis here very cheaply. Great help with house and kids.
Certificates: This is currently being covered really thoroughly in the Teacher's Tips area. The general concensus seems to be: don't bother. Not for China, anyway.
Finding Jobs: Great choices for this all over the Links page. AsiaXpat is my weapon of choice. Research is indeed very important; we'll be glad to help you get started here if you need us. Bear in mind, though...with a job in China, there will always be a point where you have to just hold your nose, dive in, and hope you made the right choice. There are definitely no guarantees here. English schools, especially private English mills, pop up and disappear again like toadstools. (Hmmm. Toadstools. Not a bad analogy...) There's no way to ever get reliable information on all of them.
There's lots of stuff here, in the Library and elsewhere, to help you make judgments about schools. And we're all just a post away if you need more.
Recruiters: Don't use them.
Don't use them.
Don't use them.
Please. Do yourself (and most of the rest of us here) a favor. Stay away from recruiters, no matter who they may be or how wonderful and helpful and sincere they may seem to be. You'll have enough to worry about without adding this completely unnecessary layer. Always deal ONLY with principals from the final employers.
Please.
Shuangyashan: I'll leave the Resident Expert to handle this one. You'll probably get quite a range of opinions on Inner Mongolia in general...personally, I'd love to travel there more but not sure I'd ever move there. I lived in an adjacent province for a year and that was close enough for me.
LT, a question: It is as you say a coal mining town, and no doubt burns a lot of coal throughout those long winters. How's the air quality there? With little-bitties, this is an important consideration...