1992 Honda Prelude Engine Problems

1992 Honda Prelude Si 2ws with H22a1 engine, Manual, 100,000+ Miles.

Hi, I live in Illinois. The weather is always cold here.
The engine of my car has several problems in the winter:

It’s hard to start the engine when the car cools down for a night. I mean I can start the car, but it takes 3-4 seconds of the ignition. I changed the spark plugs, but it wouldn’t help.

Once the cold engine started, there was a “Grinding/Scraping" sound from one of the pulleys of the engine, but it won’t happen in the warm weather. Is it the sound form the water pump? Do I need to replace it?

Once I first start my car in the cold day, the rpm goes up and down between 2000-3000 by itself. If I put my hands on the steering wheel, I can feel the engine is shaking when the rpm goes up to 3000 and then slow down when the rpm goes down. After drive the car few miles, it will back to normal. Is it the problem of the injectors? I had used some injector cleaners but didn’t help.

Sometime I turn off the car and start again, the car stalls. It looks like run out of gas or the battery is dead. But the battery is in good condition, I just can’t start the engine. If I take the key out and let it stop for a while, the car back to normal. This problem seldom happened but it also happened in the warm weather. One mechanic said that I have to replace the distributor, but I am not sure that is the problem.

Sorry that my car has a lot of problems, but I hope I can get the ideas of what happen to my car and fix it correctly.
Thanks.