If you change that diode, make sure you know what type it is, a regular rectifier, zener, whatever, don't just throw any old diode in. Try to read the number on it and look it up. Do you have a multimeter? A lot of the things in there can be easily checked with a meter.
Trying to run it off a clean 12 V source like a battery would be a good idea, like someone else suggested. In a couple of your pictures there's a 9v regulator [7809CT], and I think I saw a 5 V regulator [7805]. The 5v probably supplies the display and the 9v the radio circuitry. If you have a meter that would be a first place to check, check the three pins on each for a 12v input, a gnd, a 5v output on the 5v and a 9v output on the 9v. Bad solder connections is another good possibility.
That radio looks to be in nice shape inside. It doesn't look like anyone's worked it over, looks nice.
The back-up battery is probably just for a memory function, when you replace it make sure you use the same type, cadmium, lithium-ion, whatever it is, it should say on it what type it is and the voltage, or have a part# that can be looked up.