After careful consideration, this may be your scenario:
Bad compressor or clutch. In the car wash, the belt gets wet and because of the bad compressor, slips on the crankshaft pulley. This makes a whining noise and, because it's slipping, doesn't drive the alternator making your battery light come on. Another possibility is a bad tensioner (losing spring tension).
Inspect your belt you may see some spots where it's rubbing through. With the engine off turn the A/C clutch by hand with a 10mm wrench, it should turn easily without binding.
To check the A/C with the clutch engaged, remove the belt.
Unplug the connector from the A/C pressure switch on the accumulator and use a paperclip to short between the two pins on the connector. Turn the key to on - but DO NOT START the engine. The A/C clutch should be engaged, spin again by hand without the wrench. It should move just as easily as before.
The constantly engaging in normal operation could be a sign of low refrigerant charge or a sign of the clutch going bad. This happened to a fellow here a couple months ago. At any rate, there shouldn't be any shudders when engaged.
Other possibilities could be something freakish. Let us know how it works out.
