I do not know where the Canadian school is here in Qatar...so can't say anything about a good area t live in around there.
West Bay lagoon is very nice, but pretty expensive...unless you go to the ZigZag tower, which has come down in price, but is awful to live in from what friends told me.
There are hotels in West Bay lagoon that also have residences, quite nice from what I saw. Don't know what they would cost though.
You can get a villa for the amount available to you, but that would be unfurnished, and the compound would be kind of mid level. You would still have pool, gym, etc.
You might want to try the Al Waab area.
Now, traffic is really bad no matter where you are,so if you can find a good place to live at your budget, IMHO, take it.
I think new areas like Ain Khalid, Simaisma etc, are less expensive, but they are further out of the city.