Thanks Dean you have a very analytical approach to problems !!
"Connecting the large 'Gnd' Pin to ground and applying a voltage to all other pins at the end of the strip does nothing" yes that is true
"Connecting the large 'Gnd' pin to ground and applying a voltage to the legs of the resistors gets you green and red, is this on all 3 LEDs at once??" No light up green and red only the last led
"Connecting power to R3 (the resistor on its own) gives you green?" NO nothing happen (only the couple of the other resistor make led light up labelled 331; 271)
"Connecting power to R2 (the left resistor of the pair in your last photo, marked 331) gives you Red?" YESSS 331 red 271 green
"And R1 gives nothing where you expected blue?" Well R3 gives nothing where I expected blu while R1(271) give me green
"Check to see if pins 1,2,3 are connected" No they aren't
"Check to see if pins 7,8,9 are connected" No they aren't
"Check to see if pins 13,14,15 are connected" Yep that should be ground
"What voltage are you using when you do your test?" For my test i use a 3 volt power supply
"Try putting gnd on pin 15 and power on pin 6" Nothing happen
"Try gnd on pin 14 and power on pin 5" Nothing happen
"Try gnd on pin 13 and power on pin 4" Nothing happen
BUT.........
if I put gnd on pin 4 and power on the second half E of the connector light up blu led of the first led
if I put gnd on pin 11 and power on the second half B of the connector light up red led of the first led
if I put gnd on pin 3 and power on pin D on the first half of the connector light up green led of first led
if I put power first on pin 9 then 8 then 7 ( with gnd on GROUND bigger pin of the connector) i see last led light up in green(9) red(

blue(7)
if I put ground ( with gnd on GROUND bigger pin of the connector ) and power on pin 11, both second and third led light up red gentle not bright