How would I go about making a projectile ignore the moment when a mob turns red and goes invulnerable after being hit? This way my projectiles could hit them at a more rapid speed and still deal damage.

 

edit: If there is a way to remove the period on my mobs, that would work too.

I believe it is controlled by the hurtResistantTime field in the Entity class.  The good news is that it is public so you should be able to control it even on vanilla mobs.  It is normally controlled in the attackEntityFrom() method so you might want to check that out to understand how it normally works before messing with it.  At very least on your custom mobs you can @Override that method to get rid of the hurt resistance.