(1) was speculation, but yes, you're right, bluebrother. Even with no battery installed, a DAP should be able to turn on when connected an AC adapter... and I should've realized that since my Dell DJ's battery is now broken and I can only run it using AC power.
If it is indeed (4), is it easy to repair that kind of damage? I'm only asking because I want to be prepared for any kind of emergency of the sort.