The dogs currently all behave the same, except for who they target. Dogs behave similar to wolves in the way they navigate and roam. This page describes some of the dogs behaviors.
Taming in copious dogs is very similar to the way you fill their dog dish, you use a cooked chicken, steak, and cooked pork-chop instead of a bone. Once the dog is tamed it will follow you directly or teleport like wolves.
The dogs must have a distance smaller than 15 blocks between them. When the requirements are met, just feed them both a cookie and let the magic happen.
When a dog loses health, you can not feed them directly. You have to place down a dog dish and fill it. When there is enough food in a nearby dog dish, the dog will come over and start eating. The dog only eats if there is enough food in a dog dish to heal it completely.