Found out a little bit more about the Qatar itinerary (I now have a Qatari Visa and I have the tickets)
Basically, fly Pit->Philadephia->Heathrow->Doha, Qatar (via Bahrain, but we don't deplane in Bahrain); the return is the same, in reverse.
Heading out, 4 hour layover in Philly, 3.5 in Heathrow. Coming back 4 hour 40 minute layover in Heathrow, 4 hour 20 minute layover in Philly (some of that is important to clear customs). Total travelling time there is 24 hours, and back is about 28 hours (not including early check in or getting bags or driving to/from airports).
But hey, it sounds really amazing. I'm not sure what we'll do with a whole day there on a holy day (Friday) before our return flight. I'm trying to see if I can experience a religious service - I hear most things will be closed on that Friday so I'm not sure what else to do.