Hello Everyone!

My car is having a weird issue when it's cold. When the engine is cold my car will start kind of ruff. When its warming up I can heard the idle going up and down a couple of times. Sometimes when the engine is cold and I drive for just 2 minutes and turn off the engine, my engine won't turn back on. It will crank but it won't turn on unless I press the gas. I have already replace the battery, spark plugs and wires, fuel pump, fuel filter, air filter, and all my belts are brand new. I have done some research and I have determined that it could be something with the idle control valve. To me that will make sense since I do sometimes experience 800Rpm at red light . I plan on taking my car to the mechanic this week. I told my mechanic all the symptoms and he said it might be coolant leaking into the engine but to me that does not make sense since my coolant level is good. If anyone knows what it could please let me know.

Car info
1997 Mitsubishi eclipse rs non turbo.
miles: 170,,,,,,,,