Hi, Scotland and welcome! Fortunatley there aren't any "bad" areas of Doha, that is in terms of crime. There are older neighborhoods and areas in which men outnumber women by about 100 to 1 and your wife may not be terribly comfortable there. Many expats live in compounds which are walled condominium communities with clubhouses, pools, etc. But these tend to have three to four bedrooms and lots of kids. You will probably have to consult a realtor. If you are looking for a two-bedroom I wonder if West Bay in one of the larger serviced apartment buildings would suit you.
Mandi