When a person bumps into something or has something dropped on them and a bruise forms, does it form because of the object hitting the person or from the person exerting a force on the object to keep the object from pass through?

The bruise forms because of the force exerted on the person by the object. When an object hits you, it's obvious that the object pushes on you. But the object also pushes on you when you hit it. In fact, it's a matter of perspective which is hitting which. To a person standing next to you when you're hit by a ball, the ball hit you. To a person running along with the ball, you hit the ball. In each case, the ball pushes on you and gives you a bruise and you push on the ball, causing it to acceleration away from you.

Isn't there "some" acceleration at the very start and very end of an elevator ride? Why does one's stomach take a flop when the elevator stops and not when it starts?

Yes, there is acceleration at the start and stop of an elevator ride. As the car starts, it accelerates toward the destination and as the car starts, it accelerates in the opposite direction. Your stomach takes a flop whenever you feel particularly light, as when you are falling or otherwise accelerating downward. As you accelerate downward, your body doesn't have to support your stomach as much as normal and it feels strange. In fact, you feel somewhat weightless. You have this feeling whenever the elevator starts to move downward (and therefore accelerates downward) or stops moving upward (and there accelerates downward).

There should be four forces on an occupied seesaw, not just three, right? (two kids, one gravitational force, and one upward support force)

Yes, there are four forces. In Section 1.3, I've assumed that the seesaw had little or no weight so that it doesn't needlessly complicate the discussion. That way, the only important forces are the weights of the two kids and the support force on the pivot.

If you fire a bullet horizontally and drop an identical bull at the same moment, will they both hit the ground at the same time?

Yes. The fired bullet may travel farther, but it will fall just as quickly as the dropped bullet and they'll hit the ground at the same moment. This effect explains why you must aim above the target when shooting at something far away. The faster the bullet travels to the target, the less it will drop. An arrow travels slowly enough that it will fall a considerable distance en route. You must aim quite high when shooting an arrow.