<< Would irradiation kill anthrax spores? >>
Sure, irradiation would kill anything if strong enough and exposed long enough. That's how virtually all surgical implants are sterilized.
<< Fortunately the spores are heavier than air, so they quickly wind up on the ground. >>
While that is true, at 1 - 5 microns, anthrax spores can waft a while in air currents. Usually, though, they have a propensity to get clumped together like grapes, making them even heavier.