It does. And flying Qatar is a sound decision as far as product quality goes. The Qsuites are a very good product, the Doha transit experience is comparable to the one in Dubai, the lounges are nice. It should be easy to attain status as well. One word of caution: QR has unbundled their cheapest business class fares and traditional perks like seat selection at booking and lounge access are only available at extra cost. See here for details: Also, the cheapest "P" fares don't earn as well in AAdvantage as the more expensive ones, but better than in QR's own program and in ASMP. In AAdvantage they earn 25% less (125% of miles flown instead of 150%), in QRPC they earn 50% less (75% of miles flown), same as in ASMP.
I`d choose between either AAdvantage or Mileage Plan (Alaska), yes, with a nod towards AAdvantage if you intend to fly the cheapest fares on Qatar.
Won't help you now, but until 2021 Alaska and Emirates had a partnership and you could have credited all those EK flights to AS and used them for rewards with no surcharges...
If you wanted to maximize earning, seeing you're a family of 5 also look into programs offering point pooling (Qatar, BAEC, Royal Air Maroc.)
All oneworld programs offer opportunities for award flights without scamcharges because some member airlines don't charge any, while some (BA being the most egregious here, IB has them too but they're not as high) do. Booking an AA award with BAEC avios will not incur hefty cash surcharges in addition to the Avios because AA doesn't charge these - same as the same award redeemed with AAdvantage miles.
If you plan on using your points for domestic itineraries on AA, be aware that you will not get lounge access, neither by virtue of class of service (a few premium transcontinental flights excluded) nor by virtue of AAdvantage status. If you have oneworld status on any non-US oneworld airline , you will get lounge access on domestic itineraries.