I have igloo frf434 should green light stay lit?

The green light may have failed without affecting operation.

There are video's on google that have trouble shooting for Igloo ice maker types and for the Igloo ice company and YouTube.

What if the red light is on and the green light is off what's the problem

door seal not sealing
compressor not working
could be a start/run capacitor , low gas or a seized compressor
cal a technician for a proper diagnosis and quote

Because the ice level sensor is stuck in the full or up position. As long as it is there, the ice maker is shut off and will make no ice.

There is a probe that tells the machine it's full. Change it.

The door sometimes warps and need to be lightly twisted back to where all four corners make contact and seal.
Your unit may need to be tilt back a bit so the door wants to close y itself, the front foot pegs are adjustable and can be turned to make the front higher or lower than the back side of the cooler, gravity closes the door when the unit is leaning back.

Change the thermostat.

To be honest it sounds to me like your fridge may be ok?

To extract the heat from within the fridge interior a compressor (motorised pump) moves gas around a series of pipes around the cabinets insides. This movement of gas causes heat to be picked up and taken away externally to the outside world. When the fridge temperature reaches a predetermined level (around 5C) the compressor shuts down and all is quite. The sound of running water you hear is the coolant gas changing state from gas to liquid and running through the coolant pipes. If the fridge is doing it's job and keeping the content cool then I wouldn't worry. One of the reasons you are probably noticing this sound is because it is a table top fridge. Built in fridges make the same noises but you don't really hear them so much because of the cupboards screening the sound.