Moving to Doha
You might consider taking a monthly room rental at the Ezdan Suites and Hotel. It is a 10 - 20 minute ride to the airport (25 - 30 QAR depending on taxi metre or private taxi). Ten minute walk to the City Centre Mall that has lots of restaurants and a well stocked grocery/department store.
This place is a bit worn around the edges, but clean and, if not on the construction side, quiet. Rents go from 4500 QAR for a small, hotel type room and up. 7500 QAR will give you a studio with kitchen, dishwasher, and washer/dryer, dining room table for six, sitting area with two comfortable chairs and a table, desk, TV, and king size bed. Rooms are serviced twice a week. Price includes cable internet that is high speed enough to carry Skype except marginal after 1700 and on Friday/Saturday.
Facility includes a descent fitness area, steam room, large pool, surprisingly well stocked corner grocery store, two restaurants and a fast food take-out, and parking for one vehicle.
The allowance is 10,000 QAR for F/O and 12,000 QAR for Captain so once out of the hotel, the difference between rent and allowance accumulates in your pocket.
The Ezdan is really for temporary lodging while looking for a permanent place to stay. Remember that they will require one month for security deposit, and first month rent so you will need the cash up front. They will operate on a longer lease or month to month, your choice.
Location is quite good. Twenty minute walk to the Sheraton. Ten minutes to the Corniche. Fifteen minutes to Fudruckers. A simple solution for those arriving to have some breathing space to look around though I think living there long term might be somewhat depressing.