Sounds like a nice plan! But check into what the weather might be like in S. Africa that time of year, as it's their winter.
If points were not the issue, I'd look into flying F for all or part of the way. And I'd also compare how long it would take going from the West Coast via Asia rather than through Europe or the Middle East. I suspect it would cost you more points and time that way, but I'm not certain how much more of either it would be.
Other great options to look into would be converting your Amex and/or UR points to SQ to go trans-Pacific and then (if they fly there) on to S. Africa, but of course don't do those conversions unless it looks like there's SQ availability. The best SQ option would be in its super A380 suites that fly LAX-NRT-SIN.
Also, you can use AA points to book CX business or first tix (if available) and the redemption rates are relatively reasonable: I believe it's 62.5K per ticket one way in first class from SFO to HKG. Again, I don't know if CX then flies to S. Africa, though.
Yet another option, which involves going through the Middle East, would be Etihad, using AA miles. Their F A380 suites from JFK to AUH are reportedly the best option out there.
Now if all of this is too rich or too complicated for your blood, I certainly understand. The flight options you'd consider with your son might be less elaborate and luxurious than those I'd contemplate with my wife.
So if you just want to go with what you've already looked into, I'd suggest grabbing the LAX-DOH-etc. option you see available. (You do know that you'd be transiting through DOH to get to S. Africa, right? Your summary of the route doesn't indicate that.) You can probably tack on the SFO-LAX award flights later, or simply buy them (though if they are separate tickets, leave lots of time for connections in LAX as you wouldn't be protected for your long flights if your SFO-LAX is delayed and is a separate ticket).
Can't really comment on all the relative merits of using QR versus the SA, TK, etc. options since I don't know how much more $$ those would cost. We flew SA a number of years ago and liked it. Very roomy J seats. Having flown QR last year in J, I'd say it's even nicer than what I recall about SA. But if you can fly directly from NYC or DC to S. Africa on SA that would be quicker and easier than transiting through DOH (though the DOH transit lounge is very nice). You might look into which of your options have fully lie-flat seats in J, as that can make a big difference for sleeping on such a long trip.
I'd steer clear of Virgin. Haven't flown it, but the seat set-up is such that you and your son would be in separate pods from each other, so wouldn't get to talk or otherwise share the experience much.
Finally, as guv suggested, you might want to book separate one-way tix for outbound and return. It makes booking way in advance easier in view of the 331 day limit, and can sometimes make it easier to change flights if the need arises down the line.