Well, I don't know what exactly fixed it, but this morning during my commute to work, the squeaking/chirping stopped.
Last night, with the help of a turkey baster, I swapped out 1 quart of old, burnt fluid for Redline Synthetic Powersteering fluid. However, the problems and sounds persisted and did not seem to get any better.
Took the belt covers off, tightened the Alt/PS/Crank belt down really well. Swapped out some more fluid, and started the car up. Still squeaking/chirping.
I remembered someone saying that oil might have gotten on the belt/pulleys, so I sprayed a little Simple Green on the crank pulley, ran the car and found the problems/sounds were still present.
It was late, I still had not eaten, so I gave up and figured I'd let it sit a day and go from there.
Start the car up this morning and the squeaking/chipring was even worse. Popped the hood, looked for leaking fluid, found none, checked the belts, still tight, threw up my hands, said screw it and drove to work. Halfway through flogging my car to work 'cause I was pissed at it, I noticed the chirping/squeaking was gone.
And during the commute I was seeing more and more evidence that it might have been the belts and not the pump.
I don't know what fixed it, as I altered too many variables last night. It could be the pump is now happy with the new fluid, I'm going to continue swapping new stuff in over the course of the next few days. It might be the belts had some oil on them and the Simple Green helped get rid of that. Or they might have been too loose and needed a good tightening. :shrugs: I don't know.
Either way, if the problem persists, I'm going to swap in a spare belt I have and see what happens.