I think all the areas here you can walk through any time of the day with no problem; crime is pretty rare and, if people are to be believed, never violent crime.
Prices vary wildy depending on location, but this isn't exactly a big place! You can get a nice villa of up to 5 bedrooms for around 15K QAR, and I 'think' a similar amount would rent you an apartment at the 'The Pearl' to the north of the airport. All the restaurants there have had their alcohol licence suspended and it's not known if/when they'll get it back. Some places are closing down so its a bit of a ghost town there at the moment!! Apartments there are very, very nice though.