I used VLC and YouTube as this will show the video performance the best way. If those are not laggy, any app should work. What apps would you like to see?
Audio of the phone was on but very soft. I don't think you can hear it. Anyway, audio has some lag, it's about 500 milliseconds. I did some analysis on it and it seems to be the AudioRecord class from the Android API itself. That's on my old Nexus 6. Still have to see how it's on a more recent phone. I tested this also with Android Auto itself and Screen2Auto on a Nexus 5X , it gives about 300 milliseconds delay.
Touch is a bit laggy because I am using an accessibility service to inject touch. This allows only to inject complete gestures so the app is recording a gesture from the head unit and then playing it back on the phone. This is the only way to inject touch without root. I wanted to have this working first as most users won't have a rooted phone.
When root is available, touch events can be injected directly and there won't be hardly any delay. The plan is to include this also in the app.