2002 A4 Quattro having multiple electric problems after battery change. Here's a list:

Radio either plays fine or has extreme static
Radio backlight occasionally stays on after key is removed and will not shut off
Head lamp error message after bulb replacement and they're working fine
Fuel empty light won't go out regardless of fuel level
Intermittent misc errors such as lamp failure appear and disappear randomly

Mechanically, the car is fine otherwise

Is there some type of control module that could have fried out?