Re: Swimming Strong maybe. During the face down portion definitely, but when they turn their head the skin’s still wet, however your brain definitely knows that there’s air to breathe, probably automatically overriding that reflex.

RE: bobbing Kinda. Especially small children and toddlers are incapable of overriding that reflex, meaning they can and will “drown” when that part is submerged.
The bobbing motion is just more efficient (for 99% of untrained swimmers).
You’ll do the same when exhausted while swimming.
Your body generally isn’t buoyant enough in a normal swimming position. Which means you’ll want to keep your lungs as full as possible, resulting in quick exhale, inhale, and then recouping (and sinking) while being lighter, requiring less force to resurface.