5V on HIGH Digital Pins, Only While Using USB Power
"Arduino Pro Mini hooked up to the 5V power supply via the RAW and GND pins, but the supply is a regulated 5v wall supply. "

That is the problem. RAW pin expects more like 6V to 6.5V for the regulator to output 5V.

If you are supplying 5V, connect to the 5V, or VCC and GND pins.

If you are connecting to USB via an  FTDI Basic or equivalent, and the 5V supply at the same time, then the computer may not like having the USB 5V line back driven from an external source (mine complains about a "power surge" and shuts down the USB port). Make up a breakout cable between the FTDI and the Promini and either disconnect the 5V line, or put a diode in series so the Promini can't supply 5V back thru the FTDI and into the USB port.  That way the FTDI chip is powered from USB, but the Promini is not, and all the hardware is happy.

Make sure to use series resistors with LEDs to avoid overcurrent from the Promini output pin, which will either damage the IO pin, possibly damaging the 328P, or damaging the LED from overheating from excess current.  (5.36V - 2.2V)/.02A = 156 ohm. A 180 ohm, 220, 270, 330, some value in that range will do. 1K will probably be fine, LEDs these days can be quite bright even at lower current; 20mA can be blindingly/annoyingly bright.