Why is the outdoor air conditioner unit making a screeching/squeeling sound

If both your central air handling fan and outdoor compressor are running and your not seeing any sort of cooling after a 10-20 minutes than I'd venture a guess and say it's time to call an AC repair service.

I'm curious what you are pushing in that makes the compressor turn on? Are you somehow bridging the main capacitor? Capacitors tend to be a common point of failure on AC units and can actually be changed fairly easily. Here is a pretty good article detailing the steps to do so.

To be honest though I'd guess that the bigger problem is the fact that you can get it running and it still isn't putting out cold air. The whole situation sounds like it's low on coolant (have you had it filled in the last few years?). AC's usually have a low coolant pressure switch that will keep the unit from running to prevent damage to the compressor when there isn't enough pressure.

Call an AC service out, many will diagnose your problem free of charge and be able to give you a much better idea of your problem. Be warned though, if your system is old and uses R-22 prices have shot up in the last year.

I think you need to find your way into the heat exchanger and look for "foreign" matter to clean out.

Let me walk you through what is going on here. The initial fan noise you hear, is the draft inducer blower, and it does exactly what you suspect. It should take air from either your intake air vent, or older furnaces would use the ambient air of the room. After the ignition logic does it's safety thing, then the main burner kicks in with that big whoosh. The main blower will come on soon as well. At this point the flames are burning in a combustion chamber where the exhaust should go up your chimney or exhaust vent. You should not smell this for more than a minute if everything is drafting properly.

The heat exchanger is the above the flames, and the main blower moves return air over the other side of these channels and that is the heated air sent out the rooms.

Now when this metal heats up there may be some odors depending on how clean they are and what condition they are in. That would be in the air flow, so that your don't smell that everywhere is telling. A bad possibility would be a hole in the heat exchanger where combustion products could leak over. A CO detector is highly recommended.