I haven't been to Qatar, so I have no advice. I just wanted to say I love the idea! I went for a run during a long (6 hours) layover in Hong Kong last year. I was flying J and put my things in a locker, took a bus 15 minutes, then ran on some very hilly but beautiful scenery. Back in time to shower, eat and and have a glass of champagne before my flight
Logistics-wise, it was also very hot in HKG (and humid!!). I wore a tank top and shorts, but had a sweatshirt/jacket with pockets tied around my waist - my passport, printout of ticket and some money were inside. My phone was in an armband for running.
Please report back if you see any women out running! I have one day in Doha later this year, and wonder if I'll be confined to a hotel treadmill just because I'm female