It's not even because it's heavier, it's because it's way more dense.

Balls fall faster because planets are more attracted to objects that are spherical shaped like them.

I think the answer to this question changes based on your interpretation of 'falling faster'. I.e. whether that refers to the total time between the start and end of the fall or to the speed of the feather/ball to an outside observer.

I get that the heavier bowling ball affects the acceleration of the earth more than the lighter feather, but I don't see how that means it's falling faster as the meme is stating. The bowling ball would meet the earth first when dropped separately and from the same height because the earth is (imperceivably) accelerating toward it faster than it does the falling feather, but both the bowling ball and feather are falling at the same rate due to Earth's gravitational force.

Or am I missing something?

Great, now we need to run the experiment with the bowling ball and feather on opposite sides of the planet!