If your radio isn't turning on, as in zero display. It might be bad... Did you check the fuses under the hood??? The reason your door chime isn't working is because of the radio not having constant power or by just being defective. You will also notice your light and fuel chimes are gone, along with seatbelt and just about anything that dings.
If you go aftermarket you can buy a can-bus box that will bring those chimes back to life. Generally a pricey add-on to an aftermarket install. If you don't mind losing the chimes, enjoy the blessing and buy a nice new radio. You will be happy. 16-18watts rms, is alot smoother sounding than 7.8-9watts rms, from your factory headunit. Your factory speakers will come to life.