In order to use all three, the cat must take a Full-Attack action, which takes a Full-Round Action (Move+Standard), which usually means it may not move aside from a 5-ft. Step. That's generally a pretty big limitation on most things that rely on lots of attacks.
Of course, most cats have Pounce, which means they also get a Full-Attack at the end of a charge, but that's a special case. If the druid's cat has that, it can either 5-ft. Step and Full-Attack (nearby target) or it can Charge and Full-Attack (target at least 10 ft. away). Pounce is fairly potent.
Without Pounce, or without Charging even if you have Pounce, if you use a Move action to move, you may only use a Standard action to attack. The Attack action, taken as a Standard, is only a single attack so the cat would have to choose one claw, or the bite. Without Pounce, Charges are also just a single attack, albeit at a +2 bonus.
In any event, yes, all three are Primary attacks and get full Strength to damage.
And of course it's ridiculously powerful; that's the druid for you. The druid is one of the most powerful classes in the game, particularly at low levels, for exactly the reason you describe: the druid gets a pet that can give a fighter a run for his money, and then he gets his own turn.
Unfortunately, it's very, very difficult to address the balance problem inherent in 3.PF. You could nerf the Animal Companion and the druid would still be one of the most powerful characters because of his spellcasting. One step I do like to take, though, is to swap the Animal Companions of the druid and ranger. The druid is the far more powerful class, so the ranger could use it, plus as someone who is in theory the better melee warrior (though that's not actually true thanks to Wild Shape), it makes sense that the ranger would have the bigger Animal Companion to fight alongside.
Either, depending on how you want to purchase the item. Mithral Large Barding (it's large barding, only mithral!) is 15000 gp, while Large Mithral Barding (it's mithral barding, only large!)is 42000 gp. Their stats are otherwise identical. As a player, you should generally choose the best of a set of actions which are nonmechanically equivalent, so Mithral Large Barding is the way to go. In campaigns I run, the pointlessly more expensive versions of items are not generally available, unless I want to make a point about a region being mechanically backwards.
Thorough explanation for you confused peoples:
We start with an item, in this case Full Plate. When we consider Full Plate, there are lots of possible kinds of Full Plate to buy. We could buy spiked full plate. We could buy Large full plate. We could buy unusually shaped full plate. etc.
For the purposes of this explanation, we note that both mithral full plate and Large full plate are available for purchase, at different prices and with different stats.
Next we look mithral or Large full plate. Large full plate is still an item primarily composed of metal, and so it can benefit from being made of mithral, changing it's price and stats. Mithral full plate is still sized for a Medium creature, and so is eligible to instead be sized for a larger creature, changing its price and stats. The end result is that the stats of the two armor sets are identical, with the exception of cost.
The rules for modification of items always assume a "base item" that the augmented item is "similar to, except...". This is why the prices for such things are expressed as operators (+500, X3,-10%, etc.) rather than numbers. This presents a fundamental challenge for the application of multiple modifiers, which Pathfinder deals with by not allowing multiple modifications. All pathfinder mods of this nature apply their effects to a base item and result in a new item, which can then be further modified by using it as the input base item into another modifier. You can't apply these affects simultaneously, or both to some 'original' base item; that's not how the system works. Instead you have to calculate one modification and then use that as the base item for the other.
Armor proficiency is more want than need.
Animals are normally not proficient with armor. As per PRD Armor Proficiency feat:
Usually only light armor (leather or masterwork studded leather, mithral kikko) is worn with no proficiency, but it is up to you to decide if the benefit of having more armor outweights the cost of armor check penalties. For example, a Mithral Full Plate has armor bonus +9 and armor check penalty -3. In Pathfinder, you can spend three feats to get Light Armor Proficiency, Medium Armor Proficiency and then Heavy Armor Proficiency to limit applicability of this penalty, but you do not have to.
Masterwork cost is added after everything else.
So, for a large quadruped, the cost of armor is:
final_cost = base_cost * 4 + 150gp(this gets a bit tricky when you consider special materials, which have masterwork cost built-in, but reverse-engineering works).