You're right--it is either screen or board.
No choice but to rule out screen with a known good LCD.
I have seen this kind of thing before---but usually it would be post-drop damage or water damage. Never out of the blue.
A bad LCD will behave like this, but so will serious logic board problems. I have seen both.
The chiming is typical---iPad yelp like that when they have a backlight problem, and knock it off when the problem is fixed. Sometimes it is the device prompting you to enter 'trust this computer' on the screen.
If I were you I'd take it to a local shop and ask them to slap an iPad 3 LCD on there and see what happens.
jessa