Lights staying on after car shut off

Hello, I have been having interesting problems with my 2010 Prius 2.
First, my interior overhead light stopped working when the doors are open. It still works when the switch is flipped on, but not when any of the doors are open. This problem has remained since it started several months ago.
Next, the tail lights were staying on at about half brightness after the car was shut off. If I locked the car with the remote, the lights would stay on, but if I locked it with the key, the lights turned off. It did this a couple of times, then stopped doing it.
My latest problem started Saturday night after I came home. When I parked it, the left (driver) side turn signal lights stayed on front and back. Not blinking, just staying on. If I turn the turn signal on, it blinks slightly brighter, but the filaments are getting constant power. I pulled the fuse for the turn signal and it made no difference. The bulbs stay on. Unfortunately, this caused the 12V battery to die on the car. I jump started it today and it is running now, but the turn signal lights are still on and the tail lights were on at half brightness initially, but have gone off now. I need to drive this to work, so need to get it fixed soon.
It looks like the lighting issues may be a symptom of the 12V battery going bad, but I don't understand how a bad 12V battery will cause power to go to the turn signal or tail light bulbs with no fuse in. I am planning on replacing the 12V battery next payday, but are there any other suggestions to check in the meantime?

Thanks,
Tom

 

I have no reason to believe there are any rodents anywhere near the car. The battery voltage was 12.3 this morning when I had to jump start it to go to work. Obviously it's over 14V when running. I'm going to replace the hazard flasher to see if that's the problem. Apparently it's under the dash above the steering column, which explains why I couldn't find it before. Now I just have to find one that works with regular bulbs.

 

To be fair, the 12.3 volts was after sitting for 10 hours with 2 incandescent bulbs on. I was surprised it was that high after a steady, prolonged drain. I don't think the battery voltage being low is causing the lights to be on. The voltage is low because the lights stay on.
We got the replacement flasher relay yesterday and my wife will get it changed out to see if that fixes the problem.

 

I figured out the problem! The Curt trailer wiring box failed internally allowing +12V to (sometimes) keep the taillights on, and (sometimes) keep one or both of the turn signals on steadily. I removed the fuse to the Curt box and it now works fine. We did change the battery, though since it was 10 years old and got fully drained (down to +3.4V) again.