bushytails

Ok, so you have charged the system, and the switch still does not turn on even with pressures (high and low, since the compressor is off) over 30psi? It could be 3/8-24 thread. Grab a random bolt/nut and compare...

Battery voltage fluctuating with the turn signals is normal. The two 1156 bulbs draw about 5A when hot (~27W each), and at least double that cold, i.e. every time they blink on. The stock alternator is 55 or 60A iirc. You're pulling surges of a quarter of its high-rpm rating, and you're at idle (since you can hear the fuel pump) where it can barely keep up with the loads anyway. And when the voltage dips, the fuel pump slows down. A spark when connecting the battery is also 100% normal, and does not imply anything wrong, modified or otherwise. You're charging the capacitors in every electronic module, including the radio, ECU, fuel pump controller (I can't remember if that year has one or not), cruise control module, etc etc.

I haven't worked on one of those, but the most common cause of such problems is bad wiring between the body and the door where it flexes, followed by burnt out speakers. Does opening and closing the door, or wiggling the bundle of wires to the door, make it go on and off? Does thwapping the door over the speaker make it go on and off? Does poking something through the grille and pushing on the speaker cone make it go on and off? If it's not the door flex wiring or the speaker, then look into the radio itself and the rest of the harness wiring.

Yep. It slows the oil from draining down, so everything is oilier on startup.

Have you put gauges on the system?

A bottle of Lucas can help with startup noises, although I haven't tried it on that exact engine.

If it's 5 years old, it's probably just time for a new battery, as long as it's reading >14V with the engine running, showing the alternator is good. You can try using a non-automatic charger - one with no blinky lights or any intelligence - to un-sulfate it, by disconnecting the battery from the car and leaving it on the charger for several days. You want the car disconnected because you're trying to intentionally overcharge the battery, and don't want the car seeing high voltage. An automatic charger won't work. Has to be one that's just a transformer, rectifier, and meter.

I think you're posted in the wrong section, so you might get fewer views. I don't know where to find that specific switch, but they're fairly generic - just find one with the same thread and function, or get a new binary or trinary switch. Have you confirmed the switch is bad? They don't fail very often. I assume it's staying open even when your manifold gauges show correct pressures, and jumpering it kicks the compressor on?

That's an aftermarket anti-theft module that takes a "key" that's a little flat thing that looks like it goes to a treadmill or something. The correct repair, as you already found, is to toss it in the bin and put the wiring back to stock.

Late reply, but old gen oem oil pressure senders are bimetal type, and can't be read with a meter. Aftermarket rheostat type sending units can be read with a meter. If you have the factory sending unit, you'll need a compatible working dash gauge, or to use a mechanical gauge for testing.