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 not like taming a wolf in vanilla Minecraft. You have to first craft a Dog Biscuit, and feed it to the dog of your choice by right clicking. A name screen should pop up, and then your dog should be tame.
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.