It is possible that they are related. I have seen some weird power issues in my time. But sounds like you have the famous MB 'dead battery' problem which is actually a Main Logic Board issue. Just hope that you are covered by AppleCare or Warranty.
To test if it is the battery, take it out and run it off wall power. If it works here then it could just be a weird power issue.
I'm assuming that it still powers on OK, chimes, fans spin up, light comes on.
You say dead backlight, this means that you will have a dim screen that can still be read under the right lighting (try holding it up so that a bright light source is behind the screen, and you can see the screen where the apple logo is).
If there is /no/ picture at all, try plugging in an external screen. If you have no picture on external screen, then bad logic board (most likely). If picture on external screen and /no/ picture on internal screen, then most likely bad screen (or cable).
So worst case you could have MLB and screen fault. Best case iffy battery/power issues. Repair costs could exceed cost of a new MacBook. Good luck!