Here's a Pet Relocator service if you don't want to deal with all the hassles and research ever-changing rules. My co-workers have used them.

http://www.qatarpetrelocators.com/index2.php

To avoid stoning, etc, just live in an ex-pat area/compound. Also, when in public, please keep your dog on a leash. Some people will not know how to interact with dogs as dogs are not common here. People may act defensively and the dogs may take that as aggression.

Good luck.