Why is it that we can use energy without doing work? Where does this energy go? (you could push on a wall until your arms fell off, but you wouldn't have done any work)

When you are pushing on something without doing any work, your energy is being converted directly into thermal energy inside your body. Your muscles are inefficient and they convert food energy into thermal energy whenever they are under tension. It's like a car, which uses gasoline even when it's stopped at the light. The engine keeps running but it does no work. Similarly, if you simply burned your cereal in your breakfast bowl, you would turn its energy directly into thermal energy without doing any useful work. Your body is also able to burn up that food energy and create thermal energy, albeit a little less visibly.

Where does energy go when you try to push a heavy object and it doesn't move? Thermal energy isn't made, so why do people get tired?

While it's true that there is no thermal energy made by static friction, since the object doesn't slide, your body can still make thermal energy directly. You get tired because your muscles must turn useful food energy into thermal energy whenever they are under tension. If you are doing work, they also convert food energy into that work, but even when the aren't doing work, they still convert food energy into thermal energy.

How do compasses work?

A compass contains a magnetized needle, with a north pole at one end and a south pole at the other. Since opposite magnetic poles attract one another, the north pole of the compass is attracted toward any south poles it can find and the south pole of the compass is attracted toward any north poles it can find. The earth happens to have a strong south magnetic pole near its north geographical pole and a north magnetic pole near its south pole. As a result, compass needles turn (the experience torques) until their north magnetic pole ends are pointed northward (toward the south magnetic pole located there).

When you compute total work, do you take into account the loss of energy due to friction, so that you do more work dragging a box over sandpaper than you would if there were no friction, and assuming the mass and acceleration and distance are the same?

Yes, friction adds to the total work you must do in dragging a box across sandpaper. If you were to ask: how much work must you do on the box to slide it up the ramp, then friction certainly enters into the computation. You exert a certain force on the box as you push it up the ramp. Part of that force is used to overcome sliding friction and part to overcome gravity. Thus some of the work you do on the box is converted into thermal energy by sliding friction and so is converted into gravitational potential energy by gravity. Not surprisingly, it takes more work to slide the box up the ramp when friction is present than it would if there were no friction.

Can you explain dynamic friction?

Dynamic friction, also called sliding friction, occurs when two surfaces slip across one another. Because of the microscopic structures of those two surfaces, they push on one another in the directions that oppose their relative motions. They try to stop one another from sliding. Because they exert forces on one another and they move in the directions opposite those forces, sliding friction always does negative work on the surface (it needn't be balanced; often on surface doesn't move so it has no work done on it-the other surface moves and has negative work done on it). This work doesn't disappear from the universe: it's turned into thermal energy.