The LEDs should give you no trouble. Because of the resistor, the current is limited, and the circuit will work fine both if any LED is unfitted, or if it's shorted. If the LED worked when you tested it but stoped working when you fitted it (assuming everything else is ok) it's more likely that you inserted it backwards than that it's broken. Inserting it backwards can NOT damage the chip, however. Never insert diodes used as rectifiers the wrong way around, however.
I'm noticing a lack of solder. If the board is anywhere near decent quality, the holes will be plated through with copper, so that if you solder on one side, there should still be a connection through the hole. I'm however noticing that the pin headers are not soldered (right?) That means that there's no guarantee that all the pins will mate. You probably should solder all these in place. (I'm pretty sure you didn't solder them on the bottom as there's plastic in the way. 
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Another thing you might want to check is that the microcontroller chip is inserted the right way around. It sounds crazy, but there are images, official product images even, all over the interwebz, where the chip is inserted the wrong way around. (Knuckleheads!) The chip should be inserted so that the notch is pointing "down" i.e. NOT toward the side where the USB and power connections are.
