Case for pi and touchscreen: https://www.amazon.com/gp/aw/d/B01HV97F64/
Suction cup: https://www.amazon.com/gp/aw/d/B076XZ9JN7/
Car charger: https://www.amazon.com/gp/aw/d/B00ISGCAJM/
Might require a little finagling...
Rpi startup takes a little while, so I don't know how nice it would be to wait 1 minute for your android auto system to startup.
For the second one... you'd need to maybe consider tapping into the CAN bus and listening for a door unlock event, then powering up off that, and/or stripping everything out of the kernel and services that isn't needed... you could probably get that boot down to 10-15 seconds
The linked "features demo" video on youtube shows it booting into the Crankshaft app from power on (through Rasbian Lite) in ~11 seconds.
https://www.youtube.com/watch?v=tFEpfuDBDjM&feature=youtu.be
Even with noatime?
PS: If you really hate the bulkiness of the Pi3B, you can try the Pi A+. I haven't tried it because I don't have it, but it should also work.
*A medium amount of money