If you shot a gun and dropped a bullet at the same time, how could they land at the same time? Wouldn't the acceleration behind the bullet keep it in the air longer?

If you shot the bullet horizontally, it really would hit the ground at the same time as the bullet you simply dropped. During the firing, the bullet would accelerate like crazy, but only horizontally. It would leave the gun with a velocity that was only in the horizontal direction. With no forces pushing on it horizontally after that (we'll neglect air resistance), the bullet will make steady progress down field. But at the same time, it will begin to fall. The vertical component of its velocity will gradually increase in the downward direction as it falls. Like the dropped bullet, it will drift downward faster and faster and the two will hit the ground together.

If the downward motion of lifting a weight transfers energy to you, why does your arm get tired?

Your body is unable to store working that is done on it and also wastes energy even when it is not doing any work. When you lower a weight, the weight does transfer energy to you, but your body turns that energy into thermal energy. You get a little bit hotter. If you were made out of rubber, you might store it as elastic potential energy (like a stretched rubber band). Instead, your muscles don't save the energy in a useful form. As for getting tired, your muscles turn food energy into thermal energy even when you aren't doing work. That's what happens during isometric exercises. There's nothing you can do about it. It's like a car, which wastes energy when it's stopped at a light.

In the book, pg. 24, last paragraph you write: "By doing work on the piano, you're increasing its energy, transferring energy from it to yourself." Wouldn't the transfer be from you to the piano?

Yes, and that is how it appears in the book. The last few words there are actually "energy to it from yourself."

If Newton's third law is true - then how can you move anything? If it exerts the exact same amount of force on you that you exert on it, wouldn't the net force be zero and the object wouldn't move?

The total force on the two of you (the object you're pushing on and you yourself) would be zero, but the object would be experiencing a force and you would be experiencing a force. As a result, the object accelerates in one direction and you accelerate in the other! To see this, imaging standing on a frozen pond with a friend. If the two of you push on one another, you will both experience forces. You will push your friend away from you and your friend will push you in the opposite direction. You will both accelerate and begin to drift apart. Each of you individually will experience a net force. (It's true that the two of you together will experience zero net force, which means that as a combined object, you won't accelerate. The way this appears is that your overall center of mass won't accelerate. It will remain in the middle of the pond even as the two of you travel apart toward opposite sides of the pond.)

How do you push a shopping cart and have the cart exert the same force on you, if you are still traveling forward? Friction? Air Resistance?

When you push a shopping cart straight forward down an aisle (just making things simple here), you are pushing it forward and it is pushing you backward. If nothing else were pushing on the two of you, the cart would accelerate forward and you would accelerate backward. But the cart is experiencing friction and air resistance, both of which tend to slow it down. They are pushing the cart backward. So you must keep pushing it forward to ensure that it experiences zero net force and continues at constant forward velocity. As for you, you need a force to keep yourself heading forward; otherwise the cart's backward force on you would slow you down. So you push backward on the ground with the soles of your shoes. The ground pushes back (using friction) and propels you forward. As a result, you also experience zero net force and move forward at constant velocity.