Automatic transmission stalling? RPMs falling?

About one month ago, I bought a car (about 9 years old) that has suddenly started to stall sometimes when I slow for a curve or come to a stop at a redlight. When I try to restart it (or occasionaly when I try to start it in the morning), the engine will turn over, but the RPMs drop to nothing immediately. I can give it gas, rev the engine, quickly brake and put it in gear, and quickly give it more gas, but that doesn’t always last. If it does work the car leaps like I’m trying to start a manual in 2nd. One mechanic thought air or fuel filters were clogged, another adjusted the idle higher (which hasn’t totally solved the problem), and a relative said he’d had similar problems and had to replace the fuel pump and tank. I don’t know of anything that brought this on, other than the [apparently unrelated] fact that I ran over a wire croquet wicket in the yard the first time the car stalled. What should I look into first? (Several of the lights on the dash stay lit after the engine has died…like brake and engine light.)

I’m thinking two things. The first is that you need to find a competent mechanic.

The second is that someone ought to check your idle air control (IAC) valve. Its circuit should be checked, and often there are ways to check the valve, but you might just need a new one. At least that’s what it sounds like to me.

Working backwards, it is normal for the dash lights to light when you are stalled. A wire croquet wicket could be causing you problems, if there are parts of wicket stuck to your engine’s external belts and pulleys.
I agree that you need a fuel pump and regulator unless you find a croquet wicket wound up around your power steering pulley.