I replaced my battery a couple months ago and.

I replaced my battery a couple months ago and then it seemed the other day my alternatior tested bad after my battery light went on. We replaced that but still it will start but then it dies like the battery is gone dead and that's all my car is runnng on. A couple days before that I had to take the light bulbs out of my car inside because my car started to act like the door was still open. In fact with the door shut and the key in the ignition it still rings like the door is open and I am leaving my keys in the car. Not sure if those things are related? My automatic door locks stopped locking when I put them in gear. Even though I have a new alternator people are saying it can be bad? Really?HELP ME PLEAE