Connecting battery and LED to Arduino Due
What you want is something like this (ignore the fact that my mini breadboard is stuck to a shield):


My starter Arduino kit came with the connector for the 9 volt battery into the power socket on the Arduino, you may need to make one.  You cannot connect the 9V battery to any other part of the Due that I am aware of (without damaging it!).

It is perfectly safe to power your Arduino with 9V this way, there is circuitry to step the voltage down to the 5V and 3.3V that the board requires.

In the connector strip nearest this power socket (labelled Power on my board) there are a couple of Gnd connections and 5V and 3.3V connectors perfect for what you want to do.  I have used the 3.3V connector in my photo, but the 5V one should also be fine with the right resistor.

In your picture you don't appear to have anything connecting the LED to ground, but other than that, if you were to connect the battery up as I have shown and the red and black LED connectors to 3.3V and Gnd respectively, it should work as desired.

As a next step, you could load the sample sketch Examples->01Basic->Blink, write it to your Due, move the red LED connecting wire to pin 13 on the opposite side and your LED would flash (along with one of the LEDs on the Due board itself).

Edit (to specifically address your concerns):


The 9V battery will not fry the Due if it is connected through the power socket, which has circuitry to drop the voltage to safe levels.
The 9V battery will not damage the LED or resistor as they are connected to 5V or 3.3V that the Due outputs, not the 9V from the battery.
Only connect the battery through the power socket.  If you have a good quality steady 5V power supply, you can power the board through a 5V pin (at least I have been able to), but you bypass the voltage regulation circuitry that way and it is not recommended.  Connect the black wire from your LED to any Gnd pin, connect the red wire from the LED to 3.3V or 5V.


Additional edit:
You can connect the battery directly to the Arduino Due board without needing a plug to connect to the power socket.  Connect the red battery wire to the Vin pin and the black battery wire to a Gnd pin.