Hi @Anglo_Rat_, S10 user here with the Android 12 update.
1. In June 2020, Samsung dropped the support for MirrorLink which was implemented in the Car Mode application, available in Galaxy Store. So you won't be able to run this mode unless you try installing it unofficially from some APK mirror sites. Now you can use only Android Auto.
2. I used different USB cables and I can confirm that it doesn't need to be the one provided with the phone. What is required is that the cable must not be charging-only, as well as the USB C - USB A adapter. Do you have the Android Auto application installed from Google Play?