http://www.obr-recording.co.uk/Arduino/RGB Strip.png
In my schematic I have labelled LED pins and Resistors so if I say Pin1 that is the pin labelled as 1 on the LED etc.
I have started piecing together your information.
From what I understand:
Connecting the large 'Gnd' Pin to ground and applying a voltage to all other pins at the end of the strip does nothing.
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??
From what I can see in the pictures I am going to have a couple of guesses here, you will have to tell me yes or no, and correct when I am wrong.
Connecting power to R3 (the resistor on its own) gives you green?
Connecting power to R2 (the left resistor of the pair in your last photo, marked 331) gives you Red?
And R1 gives nothing where you expected blue?
Could you read the marking on the last resistor, I can make it out in your photos.
Could you try the following with your multi-meter if possible?
Check to see if pins 1,2,3 are connected
Check to see if pins 7,8,9 are connected
Check to see if pins 13,14,15 are connected
What voltage are you using when you do your test?
Could you try the following and let me know the result
Try putting gnd on pin 15 and power on pin 6
Try gnd on pin 14 and power on pin 5
Try gnd on pin 13 and power on pin 4
Dean