Animal Companions definitely get into some weirdness because of the frequency with which they exceed the typical Int 2 cap that grants them partial sentience, so I would always run this sort of thing past your GM just to make sure you're on the same page, but here's my thoughts:
First some relevant text from Dominate Animal
"This spell allows you to enchant the targeted animal and direct it with simple commands such as "Attack," "Run," and "Fetch." Suicidal or self-destructive commands (including an order to attack a creature two or more size categories larger than the dominated animal) are simply ignored.
Dominate animal establishes a mental link between you and the subject creature. The animal can be directed by silent mental command as long as it remains in range. You need not see the creature to control it. You do not receive direct sensory input from the creature, but you know what it is experiencing. Because you are directing the animal with your own intelligence, it may be able to undertake actions normally beyond its own comprehension. You need not concentrate exclusively on controlling the creature unless you are trying to direct it to do something it normally couldn't do. Changing your instructions or giving a dominated creature a new command is the equivalent of redirecting a spell, so it is a move action."
This seems to let them skip the Handle Animal check if using Dominate Animal. They're clearly limited in what can be done, but note that there is no provision in this spell for being forced to do things against your nature.
Now if they get hit with Dominate Monster (less likely unless you're quite high level, but still relevant)
"If you and the subject have a common language, you can generally force the subject to perform as you desire, within the limits of its abilities. If no common language exists, you can communicate only basic commands, such as "Come here," "Go there," "Fight," and "Stand still." You know what the subject is experiencing, but you do not receive direct sensory input from it, nor can it communicate with you telepathically."
Here having a linguistically talented bird will work against you, but even if there is no common language (or language at all) there are still several things that can be communicated. Since dominate monster does contain the inherent nature limit, this is where I would probably allow the exclusive training to get the extra save, though that's interpretation, and definitely not the only reading.
Lotta text. Hope that helps.
No.
Unlike charm, dominate makes the animal your meat puppet.
Dominate animal establishes a mental link between you and the subject creature. The animal can be directed by silent mental command as long as it remains in range. You need not see the creature to control it. You do not receive direct sensory input from the creature, but you know what it is experiencing. Because you are directing the animal with your own intelligence, it may be able to undertake actions normally beyond its own comprehension. You need not concentrate exclusively on controlling the creature unless you are trying to direct it to do something it normally couldn't do. Changing your instructions or giving a dominated creature a new command is the equivalent of redirecting a spell, so it is a move action.
So they only need to make it spend a move action if they want it to operate a door knob or something.
Just like dominate person works on a humanoid, so does dominate animal work on animal companions.
The exclusive command would stop charm animal or similar effects from working, but dominate is simply too powerful. If they fail the save, they are their slave.