The battery doesn't necessarily need to be in all the time, it normally does more harm than good to have your battery in all the time when your using it while plugged in.
Does the laptop boot when you have the battery in with the AC adapter plugged in? And when you say faint shadow can you still see an image on screen, which is really faint? If you can it might mean your backlight has gone on your screen. (try shinning a bright light onto the screen when it's on, and look from a slight angle to avoid the reflection)
Could you try plugging the laptop to a TV or monitor using a HDMI or VGA cable if possible?
Try cleaning out the whole laptop of any dust.