I have lived in Doha for 10 months. I have never once seen a roach in the apartment that we lived in. Moved to a new house almost 2 months ago, again haven't seen a roach! I have seen them outside on the sidewalk when walking in Doha, but other then that I have not seen any.
As for living in Doha I don't think that you will have a problem with that. It is a small city though but very modern for sure. I have no problems finding anything that I need or want, well at least not for the most part.
Good luck with the move.
Charlene