I advise you to come for a short visit 2-3 days before you take youe decision.

I think west bay is a better area to live in,and with your budget you can get a good place.

If you do not work,it is easy for you to reach any place in Doha,no need to think about time and traffic.

Do not  worry,anyway,Doha is a very quiet and nice place for a family life.