Need help designing a custom box
Discharge protection / detecting when the car is running: your problem here is not having access to the normal signals for this. There is a way it can be done: detecting whether the alternator is running (and charging the car battery) by looking at the voltages. The voltage should be about 14V while charging and 12-13V while not charging.

Depending on what your application is, you might be better off with another 12V battery and a solar charger designed for use in RVs; this is a common system design for powering e.g. small fridges.