How does residential electricity work?
When the circuit is closed, are hot 1 and neutral both "hot"? That is, are they both carrying a current of 120V that is alternating back and forth?


First, "a current of 120 V" doesn't make sense. The units of current are amperes or amps. Volts are a unit of voltage (potential difference), not of current. 

You can have a current of 1 amp, or a voltage of 120 V, but a current of 120 V is as impossible as a length of 20 pounds or a price of three meters.

The neutral wire carries current. The amount of current is determined by the load. But its potential (voltage) remains very close to ground, so if you touched it while your feet were on the ground, there wouldn't be a very large voltage applied across your body, and therefore we don't say that this line is "hot". (Don't actually try touching the neutral line because there are a variety of ways your wiring could be faulty that could make the neutral line hot and dangerous)


  When the circuit is open, there is no current flowing. So, what are the electrons in hot 1 doing? Are they still oscillating back and forth?


If there is a parasitic capacitance (there is) between the hot line and either the neutral line or the actual earth, then to a certain extent, the electrons in the hot wire will indeed flow back and forth charging and discharging this capacitive "load". 


  When the circuit is open, the neutral is not "energized" like hot 1. Why not?


The neutral line is not energized whether there is or isn't a load. Because it's tied to actual earth ground, it can't develop a potential relative to ground, and so if you touch it or connect a load between it and ground, it won't produce any current through that load. It can't produce any potential relative to ground because if there were a potential relative to ground a large current would flow (momentarily) and equalize the potentials.

When the circuit is open, a small current might flow in the neutral line due to parasitic capacitive "loads" I mentioned above.