If one were to use an electrostatic precipitator in a house full of smokers, would the smell from the cigarettes disappear as well? Why or why not? Isn't the smell/odor contained in the molecules and the molecules are contained in the smoke particles, thus removing the odor from the room?

I'm not sure what fraction of the odor of cigarette smoke is associated with the particles of smoke. An electrostatic precipitator can certainly remove most of the particles and with them, at least a good fraction of the smell. But I suspect that some of the odor is in individual molecules that are less likely to be removed from the air. They are best removed by adsorbing them (sticking them) to a surface, such as the vast surface on granules of activated charcoal. Such granules have pores that allow the molecules to touch lots of internal surface and stick there.

Why does the spark leap from the sphere to your hand rather than from your hand to the sphere? Does it have to do with electrons (negative) being the "moving" source and protons (protons) just being there to react? If yes, why are electrons the "moving" charges and protons aren't?

In fact, I'm not sure that the spark always goes from the sphere to your hand. Most of the charge that moves between your hand and the sphere is probably electrons, but there is also some charge that moves as protons, or rather as positive ions. Electrons are have very small masses and are exceptionally mobile. Protons are normal tied up inside the nuclei of atoms (except in the hydrogen nucleus, which is simply a proton) and they must drag most of the atom with them. An electron can move easily, but the only available particle that is positively charge is an atom that's missing an electron or two and is thus a positive ion. Positive ions are so much more massive than electrons that they move much less easily.

What's the difference between energy and momentum? (like in Question 4d from the problem set - is it energy or momentum that the rubber ball is unable to get rid of?)

Energy is the capacity to do work. It's an amount without a direction. Momentum is a measure of translation motion. It has both an amount and a direction. Both of these quantities is conserved, which is what makes them interesting. But they don't always go together the way you might think. A very massive object that's moving slowly may have lots of momentum but not much kinetic energy. An object with very little mass that's moving quickly may have little momentum but lots of energy. Since momentum has a direction associated with it, an object can exchange lots of momentum with something it hits and reverse its direction, while still retain virtually all of its energy.

How come the iron didn't attract toward the middle? Does the middle not have any charge?

When you bring one permanent magnet near another permanent magnet, the like poles repel and the unlike poles attract. But if you hold the north pole of one magnet near the middle of the second magnet, it will experience very little direct attraction or repulsion. Instead, it will be pulled sideways, toward the opposite pole of the second magnet. Only when the first magnet is extremely strong, much stronger than the second magnet, will the first magnet be directly attracted to the middle of the second magnet. In that case, the first magnet overwhelms the magnetization of the second magnet and the second magnet behaves like a piece of iron. When you hold a permanent magnet near iron, you temporarily magnetize the iron in a manner that we'll discuss in the section on Tape Recorders.