My son drives a 2003 Ford Focus with 183,000 miles on it. At random times when he starts the car, the engine runs very rough at idle speed but when he drives away, it runs fine until he slows for a stop sign/turn, etc. and the engine is at idle speed again. When this happens, the engine runs so roughly, that he loses vacuum to the brakes and his brakes stiffen dangerously. The car may not do this at all for a couple of weeks of driving, and when it doesn’t do it, it runs beautifully. Also, it only runs like this if it does it from start-up; it never idles roughly in the middle of a drive in which the car started normally. I have changed the spark plugs and wires, PCV valve, fuel filter, and air filter, but it still goes through this about once a week. I bought the car used just a few months ago and it had just had the valve cover gasket changed because the valve cover was leaking. I don’t think this plays a role in the above-mentioned problem but thought I would add this info if it helps.
The problem might be caused from a dirty/defective Idle Air Control valve.
The next time the engine’s idling rough at operating temperature, locate the IAC valve on the throttle body, and tap on it with a handle of a screwdriver. If doing this causes the idle speed to change replace the IAC valve.
I’m not sure if this is related, but my 2011 Ford Focus would idle rough (engine vibration), when it was cold. It would last for several miles, then quit. The dealer decided to change out the front motor mount. The rough idle (engine vibration) has quit for now - however, the weather has warmed up since the change out, so I’m not 100% convinced that it is cured, so I may have to wait till next Fall. (Oklahoma)
What I learned from the dealer is that the motor mounts are fluid filled, and can rupture / fail. Mine didn’t, but the cold weather may have made them more stiff, therefore allowing normal engine idle to vibrate in the car cab.