I have a friend that is trying to do the same as you.
Renting is expensive and if you buy a decent secondhand car you will have something to sell at the end of 12 months.
My friend is here for 2 or 3 weeks every month and has been using a driver or renting a jeep at approximatley 5000-6000 Qatar Rials per month.
I think you will need to get an International License or a short term Qatar license every time you renew you visa.
I think about 200 Rials but not sure

The problem is ownership of the car, you will need to register the car and also get insurance. I am not sure about these details.

Please if you find out any helpful details please contact me 6730727