2 bed room flat

Housing has become a major problem in Qatar and the single highest expenditure.
If you are looking for a 2-bedroom apartment then you need to budget in the QAR 7000 range. It might be possible to find one for about QAR 5500, I know someone who did, but he looked for 2 months and it's in an old building in an area full of bachelor laborers. The ideal solution is to get company provided accommodation or to have generous housing allowance that will cover and exceed the current prices as things aren't getting cheaper and there's no reason to believe they will.
Most places are accessible so you don't necessarily have to live in your work area.
It seems you have a family so would you really want to share a villa?
There's an excellent post that has approximate prices for groceries: http://www.qatarlivi...
Naturally you know you have to also budget for nappies and other child-related expenses like clothing, school, doctors...
Try to visit Doha before your move so you can get a feel for the country and see things first-hand.

Think in terms of at least QAR 5000-6000 per term per child, multiply that by 3 for your total per school year. There are annual increases of about 10% and schools like the American School and Doha College have also started charging what they call a "capital charge", an amount in excess of the fees to cover their expenses like rent for teachers accommodation, school improvements, budget deficits, etc. In reality this "capital charge" is nothing more than a rip-off but there you have it and the Ministry of Education here knows but doesn't care.
Bottom line:
If you have a family consider your accommodation and education costs very carefully and if your employer won't cover them or help to cover them and your salary is less than QAR 15000 per month then you will suffer, especially if your wife doesn't work.

There's a Lebanese school and a Jordanian one. I'm not sure about Qatari schools. Maybe someone else can enlighten us.

You're welcome.