If I have an android auto compatible car/in-dash unit, can it be controlled with a GSM tablet or is it necessary to have a full phone connected to it?
I would just give it a try. I see no reasons why Android Auto wouldn't work with a tablet. Just download the Android Auto app from APK mirror or any other source and install it on your tablet. The same works for the cell phone mentioned earlier.
My phone officially does not support Android Auto but I sideloaded the app and I can happily use it. It is just the Play Store that wants you to believe your phone is not compatible but in most cases it will work as long as your phone has Android Lollipop or Android Marshmallow.
I do think you will need a device with a working simcard however. Wifi-only is -as far as I know- not supporterd. It wouln't make sense anyway because you'd miss the most intersting features like handsfree calling.