Is there a general rule that HD always give BAB? There are certainly some exceptions: War Hulks, for instance, gain HD that don't give BAB. Those could just be examples of specific-trumps-general, but where's the general rule?

I suspect that there isn't any general rule for all hit dice. There's probably one for racial hit dice, and one for class hit dice, but the hit dice from Inspire Greatness aren't either of those.