Don't use the beep setting when the board is powered up, you will get false readings. In that mode the multimeter is actually putting a little bit of current out the red lead and seeing if it comes back on the black lead, so if there is any other electricity flowing around the copper wires, well, you get the picture. Doesnt matter what sort of load you put in the heater plug, we need to fix the drive stage first, then you can dry your hair
With the board powered up in manual mode and the heater turned off, you should read 0V between heater+ and heater- because the transistor is not allowing any current flow, and you should get 0V on the R10 resistor because the Arduino output should be off. The fact that you are getting 0.6V at the leg of R10 when the heater is turned off tells me that something is wrong before the transistor. If the Arduino output is off but you still have 0.6V at your transistor, there is indeed a short somewhere, without seeing the board myself its hard to say where.
Below is a table of what you should be reading and different times. Leave the black multimeter lead on the -12V terminal, take the readings with red lead at the following points. Basically when the Arduino tells the heater to turn on, it puts 5V out onto the shield PCB. Some of this voltage is lost across the red LED, some of it is across the R10 resistor and the remainder is across the pins of Q2. With the heater output turned off, there should be no voltage anywhere along the test path, and therefore no current flowing through the SSR drive stage, so whatever is put out on the heater+ terminal should be coming back on the heater- terminal or at least fairly close to it. The following voltages are read directly from my system which works a treat, all your readings should be similar.
How about you post some detailed pictures of your PCB so I can actually have a look, a trained eye can pick up soldering issues much better than the un-trained eye.
Measurement Point Heater OFF Heater ON
Top leg of red LED 0V 5V (Arduino output voltage)
Bottom leg of LED 0V 3V (Arduino voltage minus voltage drop of LED)
Side of R10 furthest from Q2 0V 3V (same as above as they are directly connected)
Side of R10 closest to Q2 0V 0.8V (drive voltage for 2N222A transistor)
HEATER - terminal 12V 0V (transistor should be allowing current to flow)