Again, that's MCT vs. SCT

2hrs10min, if you've to make the connection, is crazy. If you don't mind taking a later flight, no problem at all.
I've done 2hrs 30min in PEK on two tickets, in freaking winter, with baggage to recheck. Absolutely possible if you're aware of the consequences (and, obviously, should only travel alone)
And I've done some int-int connections just above 2hrs. Made them, but not with much time to spare in some cases. For someone apparently not being very known to PEK, it's rules, and maybe having some disadvantages due to his nationality, I wouldn't suggest to cut it as close as I do. With family in tow, much, much less.
On the other hand, I've found PEK staff being very nice when you admit it's your fault. 2nd hand experience but I was "live" in by SMS, late arrival at airport (int-departure) just 50min prior to departure - still got checked in (staff was just going to leave for the gate - deadline on that one was 70min) and actually the luggage was accepted too.
I think your experience in PEK was really a bad, bad one off - it certainly doesn't represent what the average experience is (which is, overall, not great, but I would transit PEK any day.. )
Back to this discussion (which I seem to have missed till today) - thanks to GinFizz it seems that somehow this connection was indeed illegal - which would make sense, as Air China was barely running late (10min - sorry, but that's not really a delay in my books) and yet he was already considered a "no show" when he got to the counter.
So, I would really check if you got sold everything on one, single ticket - as I'm not sure that should have been allowed now. Expedia is known for some "fishy" things.. maybe you just got hit by one of those. In any way, if it was a clean, single ticket, and AA sold you it, it's them taking the risk. Air China wasn't late - they could have said to wait to AA for your party of 3, something the transfer counter apparently tried, wasn't it? In fact, in my opinion, they should have. But that's probably to ground staff wanting a timely "door close" situation.
And then, they just didn't care about your case anymore. For CA, they didn't misconnect you - they got you to Beijing on time, but your connection was crazy tight, you didn't had a boarding pass (if you somehow managed to get a boarding pass at your origin, or using web-checkin (not sure if AA does have it Ex-PEK), or anything else getting you to departures, you would have probably fine. If you presented yourself at the Gate before it was closed, and they already had removed your luggage from it, they would have some things to explain.. but considering you connected from a "foreign" carrier to AA, their systems apparently were barely talking to each other..
For me, you should try to get your expenses from AA. All of them. They should have assisted you, not let you being on your own. Unfortunately, that's happening way too often, airlines trying to shift the blame. CA was kinda put in the line of fire, since it happened at their hub, but I can't really see much fault on them.
Lessons to learn:
#1 Don't book too tight connections. Definitely not when it's on different carriers, and especially not when on different alliances. Disaster waiting to happen. Unless you wouldn't mind a misconnect, actually

(Yes, those exists)
#2 Explain your situation to the right people. CA staff couldn't do much. Probably the hotel voucher was already over what they should have done for you. AA staff was the ones to hit. Or AA city office. Or anyone from AA callcenter, whoever was able to fix your issue. And definitely to rebook you for free onto the next connection out (not even necessarily on their own metal, if they wouldn't have a reasonable fast next flight) - UNLESS, like said above, the whole thing was on two tickets.. which brings us to #3
#3 If booking two tickets, be either prepared to pay a walk-up fare, or have alternate transport, or leave LONG time. Whenever I have to make a connection (but on two tickets) I'll make sure to have either a night reserve, or at least one extra flight from the same company that I could be rebooked onto should anything happen with the first one. That means 5-6 hours to transfer usually. Or even more. You often save money that way, but the risk is real.
#4 Yes, in case you know you're overstaying a Visa/allowance, or anything similar that will cause problems at immigration, use as much time possible - and as others said before, I'd try to fix it before with police/immigration. I've been to PEK checkin counters myself often way too early (for other reasons) - they're usually opening around 3.5hrs before departure for long haul departure. I've seen some crowd at that time, but even in the Y/no status queue, it wasn't more than maybe 30 pax. Which wouldn't take 2 hours to proceed. I don't know what happened with AA at PEK, but certainly their operation is really bad there from what it seems overall?
#5 Pay more, fly direct. Not really true FT spirit (It's fun to hit all lounges along the way

) - but if travelling with family, sometimes paying more for a direct flight - or at least all on the same airline, with better travel schedule etc - is making sense. Considering the difference isn't THAT much usually, spending a few hundred $/RMB more could bring much more convenience.