Brakes

The noise that you are hearing is your abs motor/pump (whatever you want to call it). It's odd that it runs when the car has been shut off. I'm sure you know how it works, the abs module regulates the brake fluid (pressure) when a "possible lock up" is detected by the pcm/ecm. That is why (with a normal situation) you slam on the brakes, the brake pedal starts pulsating and if it's in a controlled situation, you can hear the abs module working. If it's a panic situation, it's all about the pucker factor.
You could have air in the abs system, in that case, you'll need to find someone, service manual, or alldata to get the procedure for bleeding the abs unit (different vehicles have different procedures). On late model vehicles, when the abs light comes on a "problem" has been detected and the abs system shuts down, although you still have brakes. The mushy brakes could be air, or check brake fluid level, check for leaks around the master cylinder, calipers or wheel cylinders. These issues can make the red light come on.
But with the abs motor coming on with the vehicle shut off, I would check your wiring to the abs unit (shorts to ground, short to power, etc.) because it should not come on unless the pcm/ecm tells it to. You're describing several issues that could be all related to one problem and it's screwing up other things. I would start with the most simple. Fluid level and leaks. Then maybe connections. I'm not that familiar with OBD1 systems, on an OBDII system you can pull abs codes.

Maybe this can help to get a starting point on the diagnosis.
Later

Don't know if it applies to your car, but I seem to remember some of the older Fords w/ ABS had an accumulator mounted to the ABS unit. It's basically a black bulb-looking part that is pressurized. I think it was a fairly common problem for them to loose pressure over time. I think if that was the case, the accumulator could be unscrewed and replaced.