I would check the power management settings to see at what point it is set to charge the battery. The fact that yours will run on battery at all is a different set of circumstances that experienced by others.
Your display problems seems like a contact connection problem of some kind it he panel or with a wire.
You didn't included your serial number so I can't determine whether you are in or out of warranty. If in warranty I would certainly call support and get the machine serviced.
I have checked my power settings on my battery. even it's not working. I have been used my note book max with power connection only. very few no of times i will run with battery, is this makes any difference ?
Serial no : L3 - AV618 - 06/11 Type : 0689 - D9A
If am not under warraty, can you please let me know how can i renew it ?