What is an Extra?
All characters and creatures are either Extras or Wild Cards, no exceptions. Being one or the other does not determine what the character/creature can do. This dog being an Extra isn't an upgrade from a normal dog, since normal dogs are already Extras.
Being an Extra determines only how you handle its wounds (one) and rolls (no Wild Die). It doesn't have anything to do with actions available.
What is a Special Ability?
These aren't a menu of actions the creature can take—they're not actions at all. Think of Special Abilities as Edges for monsters: they are differences from the norm with fancy names. Bite gives mechanics for the dog's natural weaponry, Go for the Throat gives its normal attacks a special effect on a Raise, and Fleet-Footed changes its run die size.
Special Abilities alter a monster's statistics from the human default. But, they are not the entirety of the differences from the human baseline—the GM is expected to know what dogs can and can't do and run the game according to real-world or genre logic.
What does Beast Master do?
The Beast Master Edge does two things: it makes animals like you and makes one in particular like you so much that it becomes a loyal companion. There's nothing about training, tricks, supernatural communication, or anything else.
With a dog as the chosen companion, what you get is a loyal dog, described in the Bestiary as a breed typical of attack dogs. No special training is implied or given, so basically you've got a big cuddly Rottweiler who would kill to make you happy, but nothing more than that.
What actions can a character take?
Since the rules are generic they expect the GM to use them to create a world (believable or otherwise), and they will not tell you who can or can't take a particular action. The rules give that job to the GM.
Whether a character can do a disarm action or not depends entirely on whether they are personally capable of performing a disarm. Can an armless beggar disarm someone? Can a songbird? Those aren't rules questions, those are simply questions the GM answers as they see fit.
Can a real-world dog disarm someone?
Sure, with training. There are trained police dogs that can disarm a target.
There's no reason why you can't train an animal companion gained via Beast Master to do the same—but it does require those months and years of training, they don't come with it when they show up.