Common ground between battery and AC/DC adapter
Yes, of course it would work. If you think about it for a moment, that's exactly how it's wired when it's used in the car.

Regarding the additional issues raised in your edit:

Inside the radio, there are two kinds of electrical loads: those that get swtiched on and off with the ignition, and those that need continuous power, such as the clock and the memory for the frequency presets.

If the radio works without any power applied to the "unswitched" power input, that means there's some sort of diode (or equivalent) that powers the unswitched loads from the "switched" power input.

In a car, there are other loads on the unswitched power bus, and the radio designers would not have wanted to power those loads through the diode inside the radio, so there will also be a second diode that prevents this. (You should be able to use a multimeter to verify this.)

Therefore, if you replace the switched power with an AC power supply, there's no way that the AC power supply can try to charge the battery you have attached to the unswitched power input. Everything will be fine. In fact, if you want to use a rechargable battery, you'll have to provide an external charging path.