I'm going to go against the grain and probably tell you what you don't want to hear.  DON'T bring your dog.  

If it was a small house dog, I'd say go ahead.  But the climate is not suitable for an active dog.  Parks are limited, and they are generally NOT dog friendly.  This means walking largely on pavement and always on a lead.  Qatar is not a dog-friendly society, although locals do make some accommodation for expats.  There are some decent vets, but nothing like the choice in the UK or USA.  The classic neighborhood walk does not exist (difficult in terms of infrastructure and climate); and dogs are not welcomed at places they would be in the UK (i.e. public spaces).  Someone will invariably get on this thread and tell us how one day they took their dog to this particular park, but I am guessing that you require something more regular than that.

We left our Welsh Springer behind with some great friends.  It was one of the hardest things I did, but bringing him with us would have been solely for my benefit, not his.  In retrospect, it was also one of the best decisions I made.