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.