We are in Dubai, 11 years now. Came for 2 and still here. We are here for at least another 3 years.
On the matter of your dog, please check to see if it is on the list of allowed dogs. There was a list of dogs not allowed into the country. I will try and find the list.
There are companies that will help with bringing your dog into the country.
Schools are a priority, check if your children can get into a decent school. There are lots of schools but of varying quality. Try and get school fees included in his package, at senior level they can be about GBP 10,000 + per year and additional fees.
Once schools are sorted and you know where DH will be working, you can work out where you want to live.
We are in Mirdif, a 4 bedroomed villa with a decent garden & shared pool will be about GBP 25,000 (and more). This is considered one of the cheaper areas. On top of that you will have utilities, we pay for a 3 bedroom villa about GBP 150 per month for water & electricity. We are frugal with it and we (DH, me and DD) are out all day and we don't have a garden with grass. I know some people budget around GBP 500 per month.
There is also a 5% Municipality fee (like council tax) on the rental amount of your villa. This is split into 12 months and paid with your utility bill.
A great site to help with things is expatwoman.com.