Should my haier 1.7 compact refrigerator run

This means the heating eliments are not coming on. Usually in this case I change out the heating timer, eliments, and thermostat. All as a set.

Check your amps off your compressor. You'll also need to make sure you capactor and thermo overload is still functional.
Give up your model number and part number you replaced would make a big differance.

These are noted for bad controller boards. It is located under the back cover. Look under the cover and see if the control circuit board is blackened or smells burned. You can get parts at this link:
http://www.repairclinic.com/Haier-Parts?gclid=CMajgvvFpqYCFYdi2godeBsFmA

The noise if from a sympathetic vibration of the unit which transforms to vibrate any loose joints, parts or fittings.
You will need to listen to the noise, use your hand to hold up the plastics or the joints in the bottom section of the unit or the frame work. It will require a lot of patience to find the source of noise. Pad it up after tapping for loose noise on the rear and insulate the noise heard till you finally get rid of any vibration noise..

you hit the nail on the head there,,,its the thermostat its dead!
change it they often stick in the on poshion when they stop working to save any food inside of the fridge

this model is thermoelectric and has no gas in it.
It uses a peltier junction to provide a temp differential. the fan will run constantly, and it will cool the interior to about 35 degrees F below ambient. theoretically...

I have an Avanti mini refrig. thermoelectric having trouble getting it to cool or cold it is a model EC15W-2

My Kenmore mini Refrigerator has this same board. One capacitor has popped its top.

no 1 setting is close to zero degrees C the setting should be around 4 for normal every day use
call a technician to fix /replace the thermostat
if you continue to use a metal object ( screw driver , knife, fort etc ) when defrosting , you problems with the fridge will soon be over as you will puncture the evaporator section and that is not repairable so you will require a new unit

Sounds like a your problem may be in the thermostat. Reason I say that is because when you leave the door open it continues to run and freeze and frost is seen on the freezer walls. I just think maybe the thermostat has become less sensitive to temperature change. To test this unplug the freezer and connect the 2 wires to the temp control together. Then the refrigerator will run continuous until unplugged. Let me know, Sea Breeze