My wife and I moved from London to Dubai (I am a captain with Etihad) and we had absolutely no problems. We used a company called Petair who shipped our two cats from London Heathrow to Abu Dhabi with Etihad.
As I work for the airline, we arranged all the documentation ourselves via the Abu Dhabi government website and collected the cats at the airport. Just be aware that if you do this yourself, you may have to pay the 500 AED vets bill at the airport via an e-Dirham card. Locating this card was the hardest part of the process.
Many landlords allow pets and we had no problems finding a villa with a landlord that allowed us to do so. I can't comment on pets in an apartment.
Best of luck.