What dog breed is good for a farm and won't run off with open gates?

I would look into a guardian or herding breed such as a Great Pyrenees or other type of shepherd. Often, these dogs are bred to spend their entire time guarding flocks with little human interaction, and do well living outdoors among the herd. Speaking with local breeders in your area that raising working stock may help you get a better idea about these dogs and see if they are right for you.

For purposes of looking after children, the first breed to come to mind is Great Pyrenees. They are flock guardian dogs, are known to be loyal and protective, and are large and robust enough to live outdoors if given proper shelter. As a thick-coated breed, they do require regular brushing and grooming to keep from getting matted. German Shepherd/Alsatians are also excellent family dogs, very loyal and protective and highly intelligent. Bernese Mountain Dogs are loyal, confident and intelligent. Rottweilers are also a good breed as they are very loyal and protective. Some people are put off by their reputation but they are a product of their environment, as is any dog. As long as the children are taught to respect and properly treat a dog, there is no need to worry.

Based on your desires I would recommend a Labrador or Golden retriever. They are eager to please dogs and so easier to train than huskies or hounds and tend to stay close. They also have less of a prey drive than some of the other breeds so less of a risk to your other pets though every dog can be different, I had labs growing up with rabbits and birds and little creatures and they did not harm them.

That answer is going to depend more on the individual dog's temperament and socialization- reaching out to breeders in your area is a good idea to see what early socialization they do, if the litters are raised with other animals/dogs, etc. Any puppy, if socialized well enough, should be OK to get along with your current pets. Most active working guardian dogs also spend most of their time with the flock than with other dogs (unless they are also guarding), but the key is to lots and lots of socialization early on so that your guard dog recognizes your pets as "friend".

Small dog breeds tend to run away less then large breed, but this is a very general statement and it can't guaranty in any way that your dog will stay on your property regardless of the breed. you do have plenty of options, these are just examples: jack Russel, Pug, Maltese, Pomeranian, French Bulldog.

I couldn't quite decipher what you were asking about with the chow chow or shar pei, but can give some comments about keeping a dog outside. My experience is similar to yours in regard to allowing a dog free-range. Dogs are very territorial and do not like to wander too far outside of their home unless necessary. Ensuring the dog is desexed, has a good shelter, has adequate food and plenty of water will usually keep a dog from wandering off and getting lost. In my experience, hounds are more likely to run away than any other breeds. I think 570 acres is plenty of room for a dog and it's unlikely that they will run away as long as they have been there long enough to know it's their home. I would avoid setting the dog loose right away in a new home, because they may run off in search of their old home. I hope that answers your question well. If I did not answer one of your questions, then please post again.

The Caucasian Ovcharka has a better temperament and is more affectionate I think. It is also a little more relaxed and should make a better companion. They both are excellent livestock guardian dogs. They both need to be trained and socialized and learn the farm boundaries and who and what they are to guard. Thanks for using Petco Pet Education Center, formerly Petcoach to help you get answers to your veterinary questions.