2000 Honda Accord RPM

Hi, I have several questions that I would like to have advice on. The first thing regards my engine losing power and eventually shuts down. This only happens after I park the car and coming back later to find that the car just won't start. However, if you were to push down on the gas hard when you hear the crank and drive it around for a bit, the car will return back to normal. But the problem won't go away and when I step down hard on the gas when the problem is going on, the whole car will shake. Also this problem comes on only once in awhile like once every 2-4 weeks and the engine light doesn't come on. Another one of my problem is that when I start the car, the Rpm moves up to 1 and it slightly moves up and down not much. Whats worrying me is that I didn't step down on the gas pedal and its doing that. My last question would be that when i'm idling, I can feel the steering wheel vibrate pretty hard. When i'm driving everything is just fine. Thank You and any advice will be greatly appreciated.