
Ryze Kuja |

Action
Varies. Handling an animal is a move action, while “pushing” an animal is a full-round action. (A druid or ranger can handle her animal companion as a free action or push it as a move action.) For tasks with specific time frames noted above, you must spend half this time (at the rate of 3 hours per day per animal being handled) working toward completion of the task before you attempt the Handle Animal check. If the check fails, your attempt to teach, rear, or train the animal fails and you need not complete the teaching, rearing, or training time. If the check succeeds, you must invest the remainder of the time to complete the teaching, rearing, or training. If the time is interrupted or the task is not followed through to completion, the attempt to teach, rear, or train the animal automatically fails.
Retry?Yes, except for rearing an animal.
ModifiersLow Intelligence Non-Animals You can use this skill on a creature with an Intelligence score of 1 or 2 that is not an animal, but the DC of any such check increases by 5. Such creatures have the same limit on tricks known as animals do.
Animal Companions A druid or ranger gains a +4 circumstance bonus on Handle Animal checks involving her animal companion. In addition, a druid’s or ranger’s animal companion knows one or more bonus tricks, which don’t count against the normal limit on tricks known and don’t require any training time or Handle Animal checks to teach.Special
Untrained If you have no ranks in Handle Animal, you can use a Charisma check to handle and push domestic animals, but you can’t teach, rear, or train animals. A druid or ranger with no ranks in Handle Animal can use a Charisma check to handle and push her animal companion, but she can’t teach, rear, or train other non-domestic animals.
An animal doesn't need to be domesticated in order to teach it tricks, but you do need a minimum of 1 rank in Handle Animal though.

Mysterious Stranger |

The main difference between a wild animal and a domesticated is how the act around people. A domesticated animal will generally accept the presence of people and for the most part not attack them unless commanded. They will also not flee when people are present. Most wild animals will look and people as something to be avoided or prey.
Even if you teach a wild animal tricks, it may still be a danger to people. For example, if you have a trained wild wolf, you will probably need to issue the command stay or down a lot more often than you would with a dog. If you rear an animal so it is considered domesticated it will avoid a lot of the problems being around people.

Azothath |
it depends.... some animal breeds are rather specialized as humans have chosen traits they value over what nature would do. So you have those breeds and local 'wild' or feral breeds. Usually in or near a populated area the domestics far outnumber the locals and releases of that genetic stock into the wild modifies the local breeds...
so it depends on your GM and Where in the environment you are.
Domestic just definition wise means the animal is habituated to humans. Usually it is more than that via selective breeding.
The Game does not go into that level of detail.
In the game this topic is more about roleplaying and reaction rolls to mounts/anml companions that are normally threatening apex predators. As expected NPCs will have negative or fearful reactions. Choosing a 'Domestic' animal avoids that.
You can buy the animal off the RAW chart and get it trained. Going off/beyond that list puts you in GM territory.
Realize horses got a bit nerfed and their 'docile' was part of it.