Grab allows an immediate grapple attempt. A grapple attempt is an attack action. Ergo anything that improves your attack rolls improve your grapple attempts.
So, yes, you get that +3 on the grab attempt.
A grapple attempt is an attack action.
The attack action is a specific type of standard action. It has nothing to do with grapple or grab.
Ergo anything that improves your attack rolls improve your grapple attempts.
Correct, but only for bonuses applicable to the weapon or attack used to perform the maneuver.
Paizo Blog Combat Maneuvers and Weapon Special Features: "Disarm, sunder, and trip are normally the only kinds of combat maneuvers in which you’re actually using a weapon (natural weapons and unarmed strikes are considered weapons for this purpose) to perform the maneuver, and therefore the weapon’s bonuses (enhancement bonuses, feats such as Weapon Focus, fighter weapon training, and so on) apply to the roll."
However:
"Of course, the GM is free to rule that in certain circumstances, a creature can apply weapon bonuses for these maneuvers, such as when using a sap in a dirty trick maneuver to hit an opponent in a sensitive spot."
So if your GM rules that the claws are essential to making the grapple attempt, then you could use those bonuses.
I misspoke; A grapple check is an 'attack roll' (not action) using the CMB in place of the normal Attack Bonus (BAB + ability + size, which you note is rolled into the CMB anyway). When the Combat Maneuver is made using a weapon (which can be done with disarm, sunder, trip), then any modifiers to attack rolls made with that weapon can be utilized (such as Weapon Focus). This is why Haste will give you a +1 on Combat Maneuvers as it provides a +1 on attack rolls independent of delivery medium.
The grab is bound to specific attacks from the creature. It only activates when the creature hits with those specific attacks. It does follow that bonuses to those attacks would transfer to the grab attempt (though not to general non-grab grapple attempts), though it is not expressly described in the rules. I thought it was; Grick was right in correcting me.