When you say a house close to QP, will you be working in Ras Laffan Industrial City (about 45 min North of Doha) or in QP offices in Doha proper. Either way, if your wife is stay-at-home, you will want to consider the benefits of living in Doha. Also, if you are planning on sending your children to American School or the new Canadian School, both of these are in Doha as well. While living in the city will mean higher rent and a longer commute for you to work (if you are working in RLIC), I think the benefits for your family would far outweigh the disadvantages. I would rather have me driving 1.5 hrs daily on the desert highways than my children.
Re. wives not being allowed driving permits ... unless this has changed, your Canadian driver's license is directly transferable to a Qatari license. All you need to do is pass a simple eye exam once you have your resident's permit, then pay the transfer fee for your Qatari license. My understanding was that the changes proposed (not sure if they've been adopted or not) were for new female drivers or female drivers from countries whose licenses are not directly transferable (i.e. India, South Africa, Philipines, etc.).
As for keeping your house in Canada. If that is the ONLY tie you have to Canada, it does not necessarily mean you will be considered a resident. You can rent it at arm's length (i.e. to someone not in your family) and have an agent recover a portion (25% I believe) for Revenue Canada. However, you must break as many ties as possible (i.e. child care benefits, health care, etc.).
And remember that even though rent may be cheaper outside the city, you will end up paying for it in time and effort; and will not be able to get away with having just one vehicle, which will be an added cost in itself. Drivers here are fairly cheap, so that is always and easy option for getting around in the city. If you can find housing in a compound, this is ideal; while it may cost a little more, you will have the advantage of a pool, gym, etc. at no extra cost (i.e. you won't have to pay membership fees to private clubs, etc. though this may be included through your company). You can find some houses in compounds that would meet your requirements on the Northern outskirts of the city for about 15,000 QAR/month.
PM me if you would like more info outside this public forum.