When I put the car - 2004 RR - in reverse the light on the console blinks. I backed up to about a foot or so from my garage door, and no alarm. Do I have to get closer than that? Is there something wrong with the system?

 

the warning should start going off within about 5 feet as a beep then a steady chime as your within 2 feet.
If the light only comes on then you have a problem,
its usually a broken wire I would guess along the inside of the rear bumper.
Take the rear bumper cover off and investigate.
The main connector is fastened at the right hand side of the body this connector can sometimes not be fully seated and thus cause the issue.
Otherwise removal of the bumper cover will reveal how the wires are running and if any of them are cut or being pinched and shorting out.