2001 Mazda Protégé 2.0 LX model.
When I turn on the AC I can hear the compressor working. When the fan speed is 1, 2, or 3 the AC button is green, the compressor is working but no air is coming out of the vent.
When I turn the fan speed knob to 4 the AC button light goes out and I have air flow from the vent. The hose from the compressor is cold to the touch but the air coming out of the vent on 4 is not cold.
This sounds like a perfect reason to take your Mazda to a professional AC shop. Too many variables here to start guessing and guessing is always expensive.
I have air on all settings including windshield only on speed 4.
Sometimes, on 4, the green light on the ac button turns on and I have cold air. Other times the light does not come on and I don’t have cold air.
This applies to speed 4 only.
Still no air from speed 1,2, or 3 in any of the vent settings.
Any Mazda mechanics out there?
Most times a blower that runs only when the switch is set at maximum speed indicates a defective resistor pack. There seems to be more going on here than that. I would follow @missilemans advice
And if the resistor has gone out this often points to an aged, dragging blower motor which has been pulling excessive current. This car should also have a compressor controller module which may be affected by a failing blower motor.
I’ll take a look at my Mitsubishi schematics this evening and see if there’s anything there worth writing home about. Back later.
I ordered a resistor from RockAuto ($29.00). I’ll come back with the results.
Lost my train of thought earlier; apparently. No Mazda schematics in my stash at all.
Keep in mind that if the resistor has failed, that is usually a symptom of something rather than a cause and the cause of a failed resistor is a dragging blower.
The reason I don’t suspect the blower is I have a very high, strong air flow from setting #4. No noise either.
A dragging blower can still put out a lot of air and not make any noise at all. The only way to really know is by checking current draw with an ammeter.
I’m not saying the blower is the problem; only that it’s a suspect.
For what it’s worth a couple of years ago my wife came in and said the blower motor in our Lincoln had blipped out momentarily and came back on. The airflow was fine, there was zero noise, and no amount of tapping on the blower would cause it to hiccup.
I hooked up an ammeter and it showed that the blower motor was drawing 27 amps on the high position; just about enough to weld with… 
It was also starting to char the connector on the blower control module. (No resistor on this particular car.)
I ordered a new resistor from RockAuto on Thursday and it arrived today, Saturday. I installed it and now I have AC with correct fan speed on all four speeds. I’m a happy camper!!!