You can never go faster than once ever six seconds. That's how rounds are built. Even if the Fighter was already in combat when the Wizard cast Cat's Grace on him, it doesn't make him suddenly go faster than once every six seconds.
No it doesn't. Even if it did, you'd have to cast it before Initiative was rolled, so you'd have to guess that a fight is going to begin in the next round per level or it'd be wasted.