Annoying buzzing sound, speedometer, acceleration issues

I am new to this forum, but have owned my 1986 535i for over two years. The car has been a solid performer for me until recently. I have a few issues that I can't seem to remedy.

1. For the last few weeks, when I get into the driver's seat and insert the ignition key, an annoying "buzzing" sound begins to emit from the area of the headlight switch (lower left portion of dashboard). Behind the switch panel is some sort of black box that looks as if it has a small warning buzzer/speaker in it. This box slides into place behind the headlight switch, though I do not think it is tied in with the lights directly as it will come on at anytime when I attempt to crank my car, even during daylight hours. It almost always comes on when I first try to crank my car, and even sometimes comes on after I turn the car off. It usually fades in rather than just "come on" all of a sudden. Is this some sort of electrical warning or something? How do I test it/locate and fix the problem? I have an aftermarket stereo/speakers/crossovers/amp and sub box in this vehicle. Would this have something to do with my problem?

2. I recently had to let my 19-year-old son drive my car to work as he was having problems with his Honda. He called me on the way and said that the speedometer quit working. My speedometer is labeled as electronic, but it DOES have a needle (NOT digital). How do I trace the problem for this? I have looked for a fuse that controls it, but can't find one. Would "pegging" the speedometer damage it in any way? In order for him to do that he would have to hit about 160 mph (I have had the car at 140 with pedal to boot). He is a typical teenager obsessed with things that go fast and constantly brags to his friends about how quick my stock Bimmer is compared to all of his friends who drive souped-up Hondas and Acuras, which kind of makes me wonder if he didn't "get on it" a bit on his way to work.

3. I just recently (Tuesday) changed the oil, replaced the plugs and power steering pump (also making a horrible whining noise when you turned the wheel) on my 5-series, AND replaced the throttle position sensor (which I thought was causing the sluggish response to my accelerator), but it turns out I STILL have sluggish response. I changed the fuel filter as well. The car cranks easier than it did and idles OK, but when I accelerate up the street, it feels as if something is holding it back. If I mash the accelerator far enough, the car all of a sudden lunges as if it has been shot out of a cannon. The higher the revs on the tach, the smoother it seems to run, but at low speeds, it performs poorly. Also, when I am going down a hill and let off of the accelerator to coast (still in gear with clutch engaged), the car spits and sputters and occasionally backfires. Could my problem actually be my oxygen sensor? Could it be plug wires? As I mentioned before, the car does not run rough at idle, nor does it seem to run rough when the clutch is mashed in and the engine is reved, only when the car is put into gear and accelerated do I have this problem...and it is drinking fuel worse than it ever has.

Please help...:(

Wow.
I'd love the follow up on this. I've been having the same problems as you described.
All three.

The third, it seems to only happen when it's wet or raining and it goes away after a day or two.

You let a testosterone driven 19 year old drive your beauty? :)

Anyway, I hear a buzz/click too... not the heater valves or anything, but an electrical type under the hood. I only really hear it if I have the hood ajar. It goes away after a few moments though.

At the time, I didn't have much of a choice. My wife had just had knee surgery, and I was driving her Nissan. I had already gone to work while my son was late for work due to problems with his car. He called me at my office and said he could NOT get his little rice-burner rolling and needed a way to get to work. I was VERY apprehensive, but told him to get the spare key from Mom and get to work.

Anyway, the sound that I hear is coming from inside the dash panel, not under the hood. At first I thought it was coming from the speaker located at the corner of my window (the tweeter near the side mirror), but i quickly discovered that it was coming from within the panel.

Any ideas?

My problems occurs in any weather...as we have had very little rain here this season. I will spend quite a bit of time under the hood of my cosmos-metallic speedster this weekend and see what I can figure out. If I come up with anything, I will be sure to let you know.

By the way, nice description of your ride in your signature...cool photo. Sounds like an awesome ride.

Is that your car in the signature photo? If it is...VERY NICE...