Why does my screen power out ( all black ) in the middle of using it?

As a first step I'd remove the battery entirely in order to make sure that AC power is definitely working. If it is, and the laptop behaves the same with the battery removed, then it's fair to say it's potentially a video issue, and then I agree the next step is to try external video in order to determine whether the problem is the chip, or the screen/cabling.

The reason I suggest removing the battery is that iBooks with near-dead batteries are notorious for leaving the machine in the state you described, i.e. needing to basically shut down and start up again in order to get the machine to wake up. If the AC is being flaky, then the whole problem could be that the laptop is at the mercy of a near-dead battery.