To answer your last question first, I wouldn’t be surprised if Henry did have the book. I think it’s sort of like his talisman, he carries it everywhere.
Henry didn’t have his bag when they kidnapped him. They cut it from him. Emma picked it up in the mine so we can probably safely assume that he won’t be pulling out the book for Peter Pan.
I also didn’t see Emma or anyone else with the bag after they found it, but that doesn’t mean in their running around, Snow didn’t stuff it in her backpack that she had on before they found Henry’s bag on the mine tracks.
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Keeper of Rumplestiltskin's and Neal's spears and war paint and crystal ball.