MEA should be OK or at least better than Saudia, but I wouldn't be real happy about the idea of connecting in Beirut.

Can you get close with a carrier that interlines with someone who does fly to DOH, so that you can use SkyTeam as far as possible and get everything on a single ticket/PNR? Alternatively you could book overnight connections someplace served by Qatar rather than picking them up in the USA with an AA connecting flight, although have you thought about driving to ORD rather than starting with AA at DTW? Some ORD airport hotels have decent airport parking packages.

This could be a trip that would benefit from the expertise and help of a really good real live human travel agent who can put segments from different alliances together on the same ticket/PNR.