Loud audio at start and fob issues, related?

I have scoured the forums looking for an answer to this with no luck. I currently have two issues with my 2007 Prius and I am not sure if they are related. I bought the car new and it has not had any major problems to date.

1. The key fob sometimes works and sometimes does not. There are times when I reach for the handle and it opens. Other times I have to push the unlock button to open the door. Other times I have to manually use the key. Same for starting the car. If I ever have to use the key to manually open the door, I have to plug in the fob to start the car. I have changed the battery multiple times on both fobs. I should also point out that the plastic covering for the buttons are worn out on both keys. It only covers the unlock button.

2. The battery died on my about 4 weeks ago (forgot to turn it off...dumb) and I had to get a jump. About a week or so after that, when I start the car, all the dashboard lights turn on (including the exclamation point!) and the radio comes on full blast for about 2-3 seconds before going silent for a second then returning to a normal volume.

Any thoughts or suggestions on what to do?

 

Thanks! Is it easy enough to do on your own? I looked at a couple of videos. I have changed the headlights out, so I think I can follow directions pretty well.

 

Just ran a test. The battery was at 9.4 volts when I started it. When I turned the lights on, it went to 8.8 volts.

 

Thanks everyone. I'll try it out. If not, my local mechanic will put it in for $30.

 

Got the 12v battery replaced. Have recently been having more of the same trouble. It's not every time, but it is getting to be more frequent. I thought it had to do with not shutting off the lights or the defroster before leaving the car (both turn off when you turn the car off). My smart key is missing the cover on some buttons. Should I replace the smart key or is this still a 12v battery problem?