You do not have to program the microcontroller at all– it comes preprogrammed, and it *is* working correctly if you see one of the LEDs lighting up now and then.  (If you do have a reason to reprogram it, you’ll need an AVR ISP programmer.)

First, try to figure out what’s wrong with the other LED.  There is *very* little that can go wrong here.  The LED is connected on one side to the pin of the chip and to the resistor, and on the other to ground.    Either, the LED is backwards, or there is a missing connection at one of those points. 

Trace the wires and connections, and try to see where they go and match them to the circuit diagram.  At every connection, look to see if you can physically observe how the parts– either wires, solder joints, or copper traces on the circuit board –are connected, to make sure that the diagram is actually implemented.  (And, if you have access to a multimeter, use that to verify connections between those points.)  

If you can verify that all of the connections are correct, it could be that the LED is soldered in backwards.  You can check this by disconnecting one of its leads (desolder one of the two wires that touches the leads of the LED, and touch it one way or the other across the leads of the working LED.  That should tell you if the LED itself is good, and connected with the correct orientation.

For the servo, follow the same basic procedure, however note that there are now three wires, not two, to trace.

With regards to the other kit, with the damage to the socket, it’s not beyond repair, so long as you haven’t actually torn off any of the pins.  Strictly speaking, a socket is not needed, and you can wire the relevant pins– there are only six of them that matter –directly to the circuit board with a little extra wire if needed.