How does a power transformer even work?

With an iron core and a reasonable number of turns on the primary winding, that winding will have a significant inductance. Connected to a mains frequency source this inductance draws only a very small current when the secondary is open, so it is nowhere near a short circuit. A 12 volt secondary would have only 1/10 as many turns as the 120 v primary and, as you start drawing current A from the secondary, only A/10 will be drawn into the primary. This means the amp x turns of the two windings cancel exactly. So the primary always draws A/10 plus the small (magnetising) current it drew with the secondary open.

If a transformer were designed for say 10v primary voltage and you connected it to 120v, then you would be much closer to the "almost short circuit" situation which you depict and you will quickly burn out the primary winding.

You are correct, a transformer has multiple coils on the same core. The core has a high permeability and this makes the magnetizing inductance large. The applied voltage creates a flux in the core according to Ep=np d(flux)/dt. This same flux is applied to the secondary and the voltage created is E=ns d*flux)/dt. There is a lot more to it, but this will get you started. See: http://en.wikipedia.org/wiki/Transformer

It takes a few weeks to answer that. Here is a free course: http://www.phy.davidson.edu/instrumentation/NEETS....