Is it true that if you threw a chest pass on the cart, instead of an overhead pass, you would have been pushed farther back in the opposite direction?

Just how I threw the ball when I was sitting on the cart didn't matter. What mattered was just how hard I threw it. The more force I exert on the ball and the longer I exert that force, the more momentum I transfer to it. I chose to threw it overhead because I thought (perhaps wrongly) that I could push it harder and longer. In fact, I didn't give it as much momentum as I would have liked and didn't go backward very fast as a result. Friction in the cart quickly brought me to rest.

Can the previous question be related to the same idea in angular momentum whereas when your arms are pulled in tight you have a smaller moment of inertia? (such as in the example you showed using the rotating stool with the weights)

When I threw the ball overhead, I certainly exerted more torque on myself. The ball pushed back on me far from my center of mass and I would have begun to rotate if I weren't sitting on the cart. But the cart and the ground exerted a torque back on me and kept me from rotating. So there are two issues going on when you throw and object: transfers of momentum and angular momentum. No matter how I throw the ball, I'm going to give it momentum and receive momentum in the opposite direction myself. I'm also likely to give it angular momentum about my center of mass and receive the opposite angular momentum myself. On the cart, I quickly give that angular momentum to the ground, so you don't see it in my movement.

How can you measure weight and/or mass through distance?

I a spring scale, the distortion of the spring is proportional to how much force it is exerting. If you measure that distortion, you can determine how hard it is pulling or pushing on whatever is attached to it. If it's supporting the weight of an object, you can determine that object's weight by measuring how far the spring distorts while supporting it.

How is it that when you transfer momentum between two objects, the total momentum is 0?

When I started on the cart, with the ball in my lap, our combined momentum was 0. Neither of us (me and the ball) was moving so the product of mass times velocity was 0. But after I threw the ball, we both had momentum. My momentum was equal to my mass times my velocity, so my momentum pointed in the direction I was going. The ball also had momentum, equal to its mass times its velocity. But since it was heading in the opposite direction from me, it had the opposite momentum from me. Together, our combined momentum remained exactly 0. Since the two of us were isolated from everything around us (approximately), we could exchange momentum with our surroundings. With zero momentum to start with, we had to have zero momentum when we stopped.

When the air track was running and a large cart was colliding with a small cart, there were times when the fast moving small cart collided with a slow moving large cart and the large cart reversed directions. Why? It seems counterintuitive that the small cart could cause such a large acceleration in a large cart. (I rephrased this question somewhat)

When the small and large cart collide, the small cart experiences the most acceleration. After all, it has less mass and since both carts experience the equal forces, the lesser mass accelerates more. But since the large cart is moving very slowly, it only needs a modest change in velocity to reverse directions. The small cart really accelerates more but the large cart's reversal of directions tricks you into thinking that it underwent a larger change.

Define Torque.

Torque is the influence that causes angular acceleration. It is commonly called a twist or a spin.

Is a spinning toy top a perfect example of angular momentum?

Yes. If you spinning it about a vertical axis (so that gravity doesn't exert a torque on it about its point), it will spin at a steady angular velocity almost indefinitely. Sliding friction does slow it gradually but if the point is very sharp, sliding friction there exerts very little torque on the top about its rotational axis. Because it's unable to exert a torque on the ground, the top can't exchange angular momentum with the earth. It spins on until it slowly gets rid of its angular momentum through sliding friction and air resistance.

If you hang a weight from a scale ten feet up and the weight descends 2 feet, is the loss in gravitational potential energy equal to the elastic potential energy gained?

Not quite. When you first let go of the weight, it falls freely because the spring isn't stretched and doesn't exert any upward force on the weight. The spring won't support the weight fully until the weight has fallen 2 feet. By then, the weight has acquired a lot of kinetic energy and it overshoots the 2 foot level. The weight begins to bounce up and down around that 2 foot point and takes a while to settle down. The weight is vibrating up and down because it has too much energy at the 2 foot point. Eventually, it converts its extra energy into thermal energy and becomes motionless at the 2 foot point. At that point, it has turned exactly 1/2 of the missing gravitational potential energy into elastic potential energy and the other 1/2 into thermal energy. This 50/50 conversion is a remarkable result related to the exact proportionality between the spring's distortion and the force it exerts.

You said that when we throw something, the earth moves very slightly in the opposite direction. If all the people on earth got together in one place and all threw balls at the same time and in the same direction, would we feel the earth move?

Probably not. The earth is more than a trillion times more massive than its entire population. If we all ran as fast as we could in one direction, the earth would experience a change in velocity in the opposite direction but that change would be less than 1 trillionth of our running speed. You hair grows faster than the earth's change in velocity would be.