Have code P-0301. Engine has rough idle, like a mis . Happens at stops and backing up in park or drive.

Check engine light on? Have it scanned and if it is a p0420 code, that will a catalytic converter code. That will make it run rough and not run well. Usually the catalytic converter will be stopped up. If no engine light on, unclear of what it could be but I hope you get the issue resolved. Thanks and Jesus loves you John 3:16-17 (KJV)

if the car is loud/noisy there is a hole in the exhaust system and this will cause a rough idle because of a loss of back pressure and the vales will not open and close properly resulting in the rough idle it also will make the engine run hotter and this will cause your spark plug gaps to increase and the spark will be less than needed to ignite the fuel in the cylinder for proper combustion which will result in poor gas mileage and lack of power is the check engine light on?

The first thing to do is not set up the idle speed, but to set the Idle mixture screw to lean best idle setting. First, turn in the mixture screw until the engine dies or runs worse, then back out the screw (recommend turning ¼ to ½ turn at a time). The engine should pick up speed and begin to smooth out.

The battery has nothing to do with the idle speed

you should have someone put an obd2 scan tool and retrieve any engine codes first
most auto part stores will do this for you
then you can come back and give me the codes and i can help you better
if this was helpful please rate this question thank you

idle air control is bad

Sometimes a rough idle like that is a fuel filter. As you've already done the plugs and wires the fuel filter would be my next bet.

bmw should NOT SHAKE THE STEERING WHEEL AT ADLE. OK SOUNDS LIKE ITS RUFF UNDER LOAD LIFT THE HOOD AND CHECK FOR A HISSING SOUND IF SO IT IS VACUUM LEAK. MOST COMMON. iF TAHAT ISNT IT LETS MAKE SURE ITS IDLEING AT 800 RPM IF HIGHER PLEASE PUT ON FORM HERE AT FIX YA AND WE CAN WORK THIS OUT CHECK IDLE, CHECK ENGINE LIGHT, AND THEY ARE BAD AT NEEDING MOTOR AND TRANS MOUNTS MY BET WOULD BE ON MOUNTS

Vacume leak or bad idle motor. With the car in park and running with the hood open listen for leaks moving your hand around the different hoses looking for a change in the sounds that you hear. The idel motor also makes the car idel up to 1500 RPM or so right at start-up then idel down, If it doesn;t the idel motor is likly. Check engine light is on when running? Test for codes, Follow code test charts don't skip any steps!.

i had the same trouble, there is a rubber boot under the cover on the top of the motor, it is coming from the egr valve, takes about 5 mins to repair and less than 5 dollars.

When you have the car in gear there is a load on the engine. If you are for some reason not getting good fuel then its trying to compensate by kicking the idle up. When you are just in park there is no load so its not struggling. Regularly using a fuel system cleaner will help with rough idle, as well as hesitation. Also it might be good to check the fuel pressure regulator.