There could be several reasons for each of the issues.

Firstly, the check engine light shows up when there is an issue with the engine sensors and its starting system components.

The misfire could also be linked to an engine starting component or a fault within the combustion process.

Rough idling has a lot of causes too but with regard to the other symptoms, a common cause is a faulty ignition system component.

Is the vehicle causing any problem during ignition? If so, then we have an answer for where the issue is originating from: the ignition system