screeching, no start (long, bad news you didn't want)
Well, I'll be the devils advocate here.
The dense white smoke on startup, that clears away, sounds like a bad head gasket, or coolant leak into the combustion chambers caused by a corroded water passage in the cylinder head. In either case you're looking at, oh, maybe 1500 clams IF you can source a good used head.
The screeching is because your cylinder(s) are filled with incompressible water, thus the engine can't turn over. The screeching is the starter pinion trying desparately to engage the flywheel ring gear, which latter isn't moving. So they're grinding away at each other.
Despite your claim to only modest machanical capability, what needs to be donen to assess the situation is fairly simple. You need to remove the spark plugs and look for little beads of water near the electrode. Presence of water confirms the coolant leak into the cylinders, and that the cylinder head has to come off.
BMW has provided pretty good tools for this simple job. The tubular spark plug wrench that came with the car works well. It's cross drilled so you can turn it with the screwdriver they also provide in the trunklid toolbox.
Once the plugs are out, take the center wire out of the distributor, and stand next to the rigt fender and have someone turn the key in the ignition and crank over the engine for about five seconds. Whussssh! out from the leaking cylinder comes a mist of air and water. Maybe put a layer of newspaper on the inner fender to preserve the wet-spot evidence. If you're lucky, there will be no water in the expelled gases, suggesting that the worst has not happened. It's my bet, though, that you'll expel some water.
About all you can do wrong is no remember to mark the sparkplug wires, use masking tape and a magic marker, number then from one to six, frontn to back, so you put them back in the correct order. Don't forget the center wire on the distributor.
Now, if you get water expelled when your lovely assistant cranks the engine, the cylinder head has to come off. Likely someone on the list will have another.
I think, since you used the car very little, you'll find that one or more water passages in the head have corroded to the point where they are leaking coolant into the combustion chamber. Heck, this was the case with the last 2002 I was given. I started it up, and standing at the rear bumper, the steam/smoke was so dense I couldn't see my feet. A couple hours later, the cause was obvious: coolant passages on three of the four cylinders had corroded into the chambers.
