Things have probably changed a lot since I worked there (2000 to 2003) but you cannot bid for flights and schedules are randomly compromised. I flew very little to Europe (not sure whether it had anything to do with me being from Europe) in comparison to Asia (I did a huge amount of flights to CMB, DAC, KTM and TRV) and at many times when I actually had a flight or two to Europe they were changed in the very last minute. It happened every now and then.
There are good and bad months of rostering. As Qatarcrew states, it various. One month you could do many turnarounds with very few, if any, layovers, while other months could offer mostly layovers and not a single turnaround. It's also very common with double sectors (DOH-JED-DOH-DXB-DOH, DOH-AUH-DOH-KWI-DOH pairings).
I would assume most layovers these days are 24 hours. Sometimes shorter. Sometimes longer.
As for the passengers, I doubt very much they differ that much than from those traveling with Emirates, Etihad Airways or Gulf Air.