97 GASE Idle problem

I've got a 97 GA SE. This problem only occurs after the car has been sitting for approx 30 mins (not running). The car starts just fine. Idle sits somewhere around 750RPM give or take. After about 30secs the idle will begin to fluctuate and almost stall out. It will bounce between 0-1000rpm. If you hit the accelerator it will increase idle (3500-4000), but there is a slight delay in it returning. It pauses for a brief second around 1500rpm, and after that everything runs fine. Any ideas?
Really need some help with this one..... its bugging the hell out of me.

Ok...

First question:
Is it possible for an alternator to cause this problem by not putting out enough voltage? I noticed tonight that when the car is idling the lights are quite dim. Hit the accelerator and things brighten up considerably. I've heard these cars are VERY finicky when it comes to electrical. And what exact (RPM) should the car be idling at? Mine idles at just above 500rpm... too low maybe?

Second Question:
How exactly does the intake (throttle body) work on these cars? I know carburators have chokes to regulate the idle on cold starts.... how is this done with SFI? It almost seems to me that once the car reaches the point where it drops idle (that point where a caburated engine would kick off the choke) thats when it begins its spastic episode.

Its definitely something thats occuring when the car drops idle as mentioned in the second question. What things are going on at that point in time?
Thanks

I am not sure about the alternator being your problem. But I ran into something similar on older throttle bodied cars. The IAC or Idle air control. It is what controls the idle speed on the vehical. It might not be bad, just dirty. It is usually located right after the throttle body, held on by two screws or torx bits (might be diffrent on your car.) when you take it out it looks like a pintle (a metal rod and spring with a cone that sits into the throttle body. If it is dirty the cone might not be sitting properly in the throttle body. Clean everything out good (including the TB) with throttle body cleaner and wipe it all down good. Then see what happens!!

Thank you for the explanation. In all honesty I've had that part out and never knew what it was! I cleaned the carbon off, but the rod doesn't move in and out all that smooth. Maybe its got a flat spot or something.... and that would definitely explain the drop in idle then the idle sticking. Is there any way to test this other than swapping out a new one. Also is there a way to test the throttle position sensor?
Thank you for your response.... I hope its as simple as that.