RE: New Radio Kills Battery
First, the radio is probably wired wrong.
Second, just starting and running the engine for a few minutes will not
keep a battery charged, in fact will run it down faster unless the engine
is run for more than a few minutes.
As for the radio being lit up, go through the manual for the radio and
somewhere you will probably find that you can set the radio to not be lit
when it is turned off.  That has been my case with two different radios
I've installed in my '90.
Barring that, is the faceplate removable?  Most new radios today have that
feature.
And I'll leave it to others to advise on the correct wiring for the radio,
more knowledgeable people out there on that subject.