Let's say someone jumps off an Olympic-sized diving board into a pool below (so air resistance is negligible). As soon as they jump, they start swinging their arms. Does this help them stay upright while falling?
Yes, swinging your arms, especially in a circular motion, helps you from rotating in the air. The circular motion of your arms has angular momentum pointing out to your sides (to your left if you are swinging your arms forward, right if backwards). Because of the law of conservation of angular momentum, your arms create a sort of gyroscope. This stabilizes your body so that the direction of the angular momentum vector does not change (note: this still allows you to rotate around the axis of rotation, meaning you could hit your face or back, but this is corrected by your arm swinging due to the law of conservation of linear momentum - see below). A similar experiment is commonly done by sitting in a chair that can spin while holding a spinning wheel in front of you by its axis of rotation. If you tilt the wheel to one side, the chair will begin to rotate to make up for the change in angular momentum (you will also find that it takes a considerable amount of force to rotate the spinning wheel, more so than if it was at rest). Similarly, with your rotating arms, if you began to rotate in the air so that your side would hit the water, your body will start to rotate about the z (up-down) axis to make up for the changing angular momentum of your swinging arms. Rotating your body into such a position would require a significant torque be applied to your body, which is unlikely to happen in the scenario described.
Instinctually, we are designed to protect our face and chest, as they contain our most vital organs. If, upon jumping off a diving board, you begin to tilt forward (which is very likely because you're jumping up and forward), your first instinct is to protect the front of your body. If your face and chest begin to move forward, you correct using the law of conservation of linear momentum, pushing your arms forward to send your head and chest backwards. (note that this is also using the conservation of angular momentum, as you start to rotate forward about your center of mass, so you rotate your arms forward quickly to make the rest of your body rotate backwards. This is the same motion as the one described above, but the angular momentum conservation has a slightly different effect here)
Yes, and indeed this is how cats manage to rotate themselves during a fall so they can land on their feet - though cats move their tail as well as their legs and this option isn't open to (most) humans.
There are discussions of the physics involved in the questions Is there a way for an astronaut to rotate? and Effect of the tail of the cat in the falling cat problem.