Hi Sarah,
First of all congratulations to the offer! PDO is one of the biggest and best companies here in Oman! I can understand you have heeps of questions, trying to answer a few.
1. Yes, there are vets, I personally only know of one veterinary clinic, they are superb though, and can handle any issue regarding relocating and transporting the animals and deliver a very personal service, so your 4-legged members of the family are well-looked after! Here is their website, you might want to contact them ahead regarding requirements for bringing the animals in the country: muscatvets.com/
2. I don't know anything about PDO accomodation apart from that they are usually of the highest standard, close to PDO and in one of the best locations of Muscat!
3. Your son might well be eligible for a family visa, if not, he can get a tourist visa on arrival (20 O.R. for 1 month). There are no universities in Oman or anywhere in the Gulf that meet European Academic Standards, so completing his studies here, I would not advise.
4. From all I've heard, British School and ABA (American-British School) have very high standards. A former colleague of mine arrived with her 10 year old son, who had big trouble integrating at school and ABA really looked after him. After 2 years in the country they had to move back to the U.K. and were most upset about having to take their son out of ABA, as they feared they would not find such a nice school in the U.K. But best try and speak to other woman with kids at schooling age.
5. You finding work depends on your qualifications and profession. Beware that if you are on a spouse visa, you are not allowed to work and working despite being on a spouse visa can have consequences for you, your husband and his company. So, if you want to work, you need find a company that is willing to put you on their own working visa. I have heard of ways around the spouse visa rule for some part-time work, but have no details on it. PDO surely has excellent PRO's who will be able to help you with the issue. What is your profession if you don't mind me asking?
PDO also has its own club with tons of events and activities for children and adults alike. In general, Oman is a much friendlier place than Qatar, and locals and expats mix more than in other gulf countries, especially at PDO.
Any more questions, feel free to ask! I can onyl recommend Oman as a country, especially for families, it is very safe and relaxed.