Which amps need 12 volts DC, and which 110 volts AC?

Car amps need 12vdc and home amps need 110vac.

If you have a speaker of 4 ohms, and you gradually turn the volume up, the
current through the speakers would presumably increase, and the voltage
would adjust. Would it?

No, the voltage at the output is (relatively) independent of the load
presented. Decrease the impedance of the load and you'll increase the
current (and thus the power) delivered to the load.

What would happen if u used a hi-fi amp to drive car speakers then?

It will run car speakers fine. You said it was stable at 4 ohms.