Scott T,
I would apply and see how it goes. QR is notoriously BAD in terms of their recruiting/interviewing/training queue processes - but you won't know unless you apply. Tell them you expect to have 1,000 hours by a certain date and then update your application accordingly if that is possible.
I don't work there, but I would say the ME is certainly where you will find the non-BA/VS widebody opportunities going forward. The other non-EK UAE carrier might be another good bet if you are looking for a widebody seat - I believe that carrier has 41 787-900 firm orders to start with first delivery in latter 2014 or early 2015.
Good luck!