2004 Prius 12 Volt Battery Questions/Troubleshooting

I have a few questions about my 2004 Prius and its 12 Volt Battery, but first here is some context to help everyone understand the situation.

Yesterday afternoon I turned on my Prius and found that only the dashboard would come on. Among other things, it showed the warning triangle, the check engine light, and that one of my doors was open (although everything was closed). I tried to reboot it, which caused everything to die. The only thing that will turn on now is the check engine light.

My hypothesis: My trunk is broken so it can't be opened from the inside or the outside. Occasionally it will come loose and the car will consider it to be opened. Usually all I have to do is push it back into place to fix that. In fact, the last time I used my car, I had to do exactly that. That was about 3 days ago. I'm guessing that the trunk became loose again and drained the car's 12 volt battery.

What I Know or Have Already Tried:
1. The main battery must be fine. I replaced it about a year and a half ago. I don't know if the 12 Volt has ever been replaced.
2. Everything is connected. I checked the connections to the 12 volt and the main battery.
3. When I try to turn on my car, a soft and steady clicking noise can be heard coming from beneath the hood
4. I tried jump-starting it. All of the electronics came on, but the engine still would not start. As soon as I disconnected the two cars, the Prius would immediately die again.
5. After the car died, the key fob won't lock or unlock anything anymore.
6. It has almost a full tank of gas, and it has appropriate oil levels.

Questions:
A. Would my 12 volt battery stop my engine from starting? I've searched online for hours and I can't get a clear answer, even on Prius chat. Some people say that you can drive the car even when the 12 volt is completely dead, and others say that it has to have some charge to start the car.

B. How much should it cost to have it replaced and installed? Most of the posts I've found about this issue list outdated prices. From what I've found online, replacement batteries are roughly $250. I've called two Toyota Dealerships, and one listed the cost of replacing and installing it at $300, and the other at $350. Is that reasonable?

C. If it isn't the 12 volt battery, what else could cause these issues? I've seen some posts saying that there could be something wrong with the accelerator or the starter, but they don't really elaborate on that. In the days leading up to my Prius dying, I didn't notice any issues. There weren't any noises, it accelerated normally, and the lights weren't dim.

D. Is there a way for me to run my car long enough to get it to a dealership or an auto-shop? This is my first car, and it's hybrid, so no one in my family has clue on how to get this to work. I can't tell if the battery won't hold a charge at all, or if I didn't charge it long enough? Towing on top of the other expenses would be very difficult on my family right now.

Thank you for your time and help!

 

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Yeah, it's pretty concerning. The engine did not spin at all. Far as I could tell, the engine wasn't getting any signal to start at all.

I just went into the car and removed the 12 volt battery. All of the white plugs were in there very tight. Perhaps there is something wrong with the wiring?

 

UPDATE: I removed the 12v battery and I took it to a nearby Autozone. Apparently it is only 2 years old, but it had only 3v in it. They are attempting to charge it now. I've read that these batteries should last for 5 years, so perhaps it's faulty? If it DOES charge, I'll attempt to reinstall it and hope that it will be enough to start the engine. Then I can get it to an auto shop.