Gravity accelerates every object at the same rate regardless of mass — that's the whole point of Galileo's (probably apocryphal) Pisa experiment and Apollo 15's very real one. What actually makes a feather fall slower than a hammer is air resistance, and air resistance depends on shape and mass in a way gravity doesn't: it scales with cross-sectional area, but the object's inertia (its resistance to being slowed down) scales with mass. A feather has enormous area for its tiny mass, so drag dominates almost immediately; a hammer has so much mass relative to its area that drag barely matters over a short drop.
Drop a hammer and a feather together from 1.6 m in ordinary air and the hammer lands first, roughly 0.4 seconds sooner in this model. Remove the air — either on the Moon, like Apollo 15 actually did, or in an Earth vacuum chamber — and they land at exactly the same instant, down to the physics engine's numerical precision.
Almost certainly not literally — it's a story told by his student Viviani, decades later, with no confirmation in Galileo's own writing. What's well documented is Galileo's careful reasoning (and rolling-ball experiments on inclined planes) that falling bodies accelerate independent of mass, which is the actual physics this tool demonstrates.
Yes — Commander David Scott dropped a geology hammer and a falcon feather on live television on August 2, 1971, and they landed together, exactly as Galileo's physics predicted. It remains one of the most direct public demonstrations of the equivalence principle ever performed.
Faster motion pushes more air out of the way per second and pushes it away faster — both effects scale with speed, so the combined drag force scales with speed squared. It's why terminal velocity exists at all: drag grows quadratically until it exactly cancels gravity.
Mars's atmosphere is roughly 1.6% as dense as Earth's at the surface — present enough for dust storms, far too thin to meaningfully slow a dense object over a short drop. It's a genuine middle ground between Earth's air and the Moon's true vacuum.