Do I need to worry about air bubbles in the first coats of drywall mud?
The purpose of sanding (or scraping) between coats is to knock off the high points. NOT bonding.
Mud (compound) fills the low points, but it shrinks, requiring multiple coats.
The goal is to have something close to a flat surface, which you get most easily by removing the high points and filling the low points.
If you don't remove the high points, you end up filling a larger area with more mud since the knife rides on the high points. More mud shrinks more, so there's more remaining low spot to fill with the next coat. If you knock off the high points, you are just filling the low points, and the amount you add each coat is less, so it shrinks less.