Yes, I think women can ride motorcycles in Qatar.
Check with any driving school around Doha in order to acquire the license. And if you have a western driving license, you can go directly to the traffic police to convert it right away. It will take you 30 minutes.

Prices are cheaper than in Europe and more or less like in the states.
A good 600cc bike will cost you from 28000QR to 50000 depends on the make and model.

Most of the bikes around for sale are either racing or off road but there are a few nice scooters and street city bikes on the market

For Example: A Ducati Monster = 40000
A Triumph 675 = 35000
A Harley Davidson Sportster or Nightster = 42000

If you don't like the models available on the market, you can always go to the dealer and order the model you want, just like I did... it will take a couple of months to arrive but it might be worth it.

2nd hand bikes are quite cheap too if you want to consider this option but you need more time to bump into the bike that suits your requirements.

The weather in Qatar is good for riding for 8 months per year... forget about riding in June July August and September.

As for traffic, the bike is the most wonderful thing that happened to me. It saves me PLENTY of time and stress. I personally haven't had any problems from the way people drive here. As long as you have a fast and light bike and know how to ride in a city.

If you need future help, don't hesitate to contact me.

Good Luck.