Considering that with all information so far collected, it looks VERY much like it was a combined OH-OH by Expedia and AA (they can internally discuss who is to pay up for it) I would settle for no less than a refund of ALL costs that you had by them. Plus, actually, a proper "we're sorry we fu... up" voucher..

Honestly, if they're selling you a illegal ticket, and then refuse to support you, it's a BIG problem. It looks like they've "learned" from it, as any such connection isn't available anymore, but it doesn't matter. They let you fly an illegal connection, it went South, and they didn't took responsibility to help you it, in fact they tried to scam you hard, and forced you to take matters into your own hands. While sometimes this can be needed (and you apparently didn't so bad yourself later, getting yourself and your family home) at the time, the follow-up events should quite clearly be that they're paying up for all (reasonable) expenses you had.

Which would be in this case here: AA "change fees", new flight home (UA), any meals, accommodation (whatever not paid by CA already - which seems to be a victim to AA and Expedia just as yourself too) and transport in Beijing, and as well the "overstay" fee you had to pay (the last one we could discuss if they need to pay it, but in the bigger context, the amount is small compared to the other costs anyway)

That said, I would expect no less than 1000$ voucher by AA (or Expedia) per person on top. IF they indeed sold you an illegal connection and let you down.

I would try to bring your case up to (higher) management level, ie escalate it.