How to determine the operating voltage and current of a laser diode?
Please forget about the specific circuit and address the main question How to determine experimentally the threshold current of an unknown laser diode?



Get the datasheet. If that's not possible, get the datasheet of a "similar" part. This will at least give you a starting point for the device characteristics.
Figure out which pins are the anode and cathode of the laser diode. This will be easier if you have completed step 1 successfully.
Connect the laser to a variable current supply. Ramp up the current slowly until it turns on. Knowing what supply to use and what range to adjust it over will be easier if you have completed step 1 successfully.