By all means use whichever airline website that provides you with the best routing/pricing outcome. Simply put your airline and frequent flyer number in place as part of the ticketing process.
Having said that you need to have completed your homework on what fare classes (in economy) will properly credit to your desired program. There are a couple of serious "gotchas" (eg. anything below H class on CX does not earn on AA) As well google flights might very well present alternative flight options that you might not have considered. Consider starting from ORD and paying for DSM-ORD separately.
I don't remember any serious issues with QR when I flew DOH-HKG in the Y cabin.... I'll have more experience with QR's Y cabin in a couple of weeks. :-(
Happy wandering
Fred