What are the physics that makes my propane tank freeze with high output?

Your tank freezes due to the same principle that makes your refrigerator or air conditioner work. When gas under pressure is allowed through a small orifice the gas cools upon expansion. This draws heat energy from the tank and tubing eventually cooling it down to the freezing point.

I don’t know which gas law that this reflects. Perhaps it is some combination of certain ideal gas laws.

I believe that it is the Bernoulli Principle at work. The short version is that, all else being equal, an increase in flow speed results in a drop in pressure and/or temperature (often both).

BTW, they make dry ice the same way, except they use CO2 instead of propane ;)

The answer is in the ideal gas equation stated in your question: Liquid propane is in equilibrium with propane vapor inside the tank. When vapor flows through an open valve, the drop in pressure (at relatively constant volume) must be balanced by a proportionate drop in temperature. Alternatively, think of the molecules of propane that escape the tank through the pipes: they have higher kinetic energy than the rest of the propane molecules left behind in the tank. Temperature is average kinetic energy, so the tank cools as the most energetic molecules leave.

The same thing happens with those cans of compressed air dust sprays that use a liquid propellant with a high vapor pressure—just like the propane in your tank.

Adiabatic expansion.

The other answers are close, but the issue is evaporation from the liquid propane in the tank in order to make up for the loss of pressure of the vapor. When a liquid evaporates it uses heat from the surroundings to drive molecules from the liquid state into the gaseous state. Each molecule leaving the liquid as a gas removes a tiny amount of heat. The liquid propane can’t absorb heat fast enough from the surrounding air, so it continues to cool until it can’t evaporate as fast as the vapor escapes throught the valve. At this point the flow of vapor stops or gets so low the attached regulator shuts down for safety. Water vapor in the air can condense on the outside of the tank and even freeze. If the tank is set in a bucket of water, the greater heat storage capacity of the water will supply a higher heat flow to the propane and it will take much longer to freeze.