Yes, you should have a current limiting resistor in series with the LED. Without a current limiting resistor you can potentially burn-up the op-amp and/or the LED. (500-1000 Ohms if you don't know how to calculate it.)
The led in the circuit currently switches at 9v. I have no clue how to figure out why.
Maybe something is wired wrong. Maybe that's not a 5.1V Zener. Maybe the pot is wired wrong. With the 5V Zener it should be adjustable from 5.1V down to about zero.
This is a comparator circuit. When the (+) input is higher than (or more positive than) the (-) input, the op-amp output goes positive. When the (+) input is lower than the (-) input the output goes negative (or to ground if there is no negative power supply).
The Zener provides a reference voltage for the comparator.
Your LED is wired so it's off when 12V comes-out of the op-amp. (The LED will be off whenever the (+) input is higher than the (-) input.) The LED will be on when the (+) input is lower than the (-) input.
An open-loop (no feedback) op-amp has (theoretically) infinite gain (infinite amplification) and it amplifies the difference between the (+) and (-) inputs.
When the (+) input is higher than (or more positive than) the (-) input the voltage gets amplified/multiplied and the output tries to go + infinity. But, it can only go as high as the power supply so if you've got a 12"V power supply it goes to about 12V. (In the real world it may only go to 10 or 11V.)
When the (+) input is lower than the (-) input the negative voltage gets amplified and the output tries to go - infinity. If there is no negative power supply, it can't go negative at all and it simply goes as close to ground as it can.
The pot allows you to lower the voltage into the (+) input. In the original circuit with the 9V Zener, adjusting the pot to the mid point would give you 6V from a 12V battery. That's less than 9V and the LED should turn on. (Of course with a 5V zener the LED would still be off.)
This is fine, but when I do this, I always use precision zeners, like the BZT585B series. I await the debate
There's a pot in the circuit so a precision zener doesn't make any difference.