Weird happenings & 12V Battery

This is my first time posting here. Can't seem to get into the Yahoo 2G forum anymore. Here is my problem and just want to see if I made the right decision. I have a 2004 with smart entry and the original 12v battery (from 2003).

Saturday: tried to lock via door button (and hatch button). Didn't hear the normal beep, just a muted click somewhere. Finally left the car unlocked and went in the store. Came back and operated the hatch and the beep sounded (apparently it was locked). Tried to start and noticed it wouldn't go into gear. Then I noticed the fuel indicator was flashing one bar (I had four when I went in). Also noticed the MFD had reset the mileage to zero. Powered down and rebooted and got a ready light. Was able to drive normally after two other starts.

Sunday: left the car in the garage and didn't start it. Was planning to take it in for service Monday.

Monday morning: I had car locked and it unlocked normally. Pushed the start and got a blank dash-no lights at all. Repeated start button pushes brought various results including a check engine light, a lit MFD with the red exclamation and a park button malfunction message, and the red Triangle. Finally after several button pushes, I got a ready light (and the red triangle). MFD was back to normal. Decided to drive it to the dealer and drop off the fob in the mail slot. Ran the car in the lot to keep the heat on, while I waited for my boss to come pick me up. ICE ran almost constantly and only shut down twice. Called the dealer later and they couldn't find a problem with the battery, but they did mention a code relating to the transmission. They offered to keep the car overnight to try and repeat the problem. I told them to just replace the battery. I'm hoping this solves the problems. I'm picking up the car tonight. I don't usually disagree with what the dealer says. Does it sound like I made the right decision?

Opinions welcome.

 

Quick update. Battery replaced. Radio presets corrected. All is right with the world. By the way, the code returned was 3102, if that makes sense to anyone. Thanks for your comments.