Your rough idle sounds to me like your air fuel mixture equipment has an issue somewhere. Could be your mass air flow sensor, knock sensor, throttle position sensor, purge valve, on and on ad nauseum. I had the same thing happen to me iny '13 1.8. When I would come to a stop, the car seem to want to surge, release, surge, release over and over. I put a quart of transmission fluid in it and it stopped surging. I still have my at idle on startup problem to replace the purge valve sensor today. Already replaced the knock sensor, and this is a pretty cheap second step for me, hoping I don't have to replace the mass air flow sensor as those bad boys are pretty expensive. Hope this helped some, and good luck.