I'll let others or other existing threads deal with the power adapter issues, but the answer here is "yes." You can use it on battery exactly as you can use it on battery on the ground. On short flights, that works fine. On a six-hour transcon it's nice not to have to worry about power conservation, shutting it off during meals, and all that.

In the days before airplane power ports people used to carry a second battery.

BTW, if you wait for a flight with nobody in front of you, you'll wait a long time unless you're on an unpopular route or like middle seats! (You can generally tell when the seat in front is about to recline or starting to. There may be the odd sadist out there who slams the seatback back full-force with no warning, but I have yet to encounter him/her in front of me.)