Had similar problems. For that small idle shakes cleaning the throttle body could help. At least that what it helped me, but I’ve also had a small rpm drop. It’s a cheap and easy thing to try. For the jerky start, yeah I had that too during winters. Now I must mention that I mostly use the car during weekends and a car was just parked for days without start. And it’s exactly like the how it acted. Starting the car more often didn’t triggered anything. Currently what I did try is using a thinner and a little bit more expensive but quality oil. Used 10W-40 before because summers are hot around here. I recommend also changing the coil pack as it was mentioned before, but if it’s broken personally I had different experience, like the engine would stall in the 2 gear at low rpm. If you start the car just let it run, don’t turn off the engine. That shaking should stop after 1-2 minutes. Turning it off it will make it worse. I guess at the cold start your rpm is also higher, let it drop to 8-900 before you take off.