do the electrons in, say, a battery, actually leave the battery, enter the circuit, travel around it, and back into the battery again (and so on, and so on...)

That is a fairly general description of what happens. What happens when electrons leave and return to the battery is a little more involved. Have a look at this and it may make some sense...

http://en.wikipedia.org/wiki/Galvanic_cell

The actual charge motion imparted by the battery is quite slow, however.

The actual charge motion imparted by the battery is quite slow, however.

The drift rate of electrons in a typical wire driven by a typical battery is very, very slow -- on the order of millimeters per second. The signal (e.g., closing the circuit by connecting both ends of a wire to a battery) travels very quickly. This signal velocity is a significant fraction of the speed of light (1/3 c, IIRC) as compared to the information velocity, or about 10¹¹ millimeters per second.

 

The free electrons in a metal act a lot like a gas. One way to envision what is happening is to construct a mechanical analog of an electronic circuit. Envision a simple electronic circuit comprising a battery, a variable resistor, and some pieces of wire connecting the battery and resistor to form a simple closed circuit.

 

A mechanical analog uses a pressurized gas tank (just pressurized air) in lieu of the battery, an adjustable valve in lieu of the variable resistor, and plumbing in lieu of the wires. Rather than closing the circuit the valve in this mechanical analog simply vents gas to the air.

 

With the valve initially closed, the gas in the plumbing will be at the tank pressure. When the valve is opened slightly, gas molecules will start leaving the valve and other gas molecules will leave the tank and start moving toward the valve. These molecules will move through the plumbing much, much slower than the signal velocity, which is the speed of sound in this case.