Jacob doesn't seem to have any guilt or hesitation about betraying Emma's trust and looking into her private things. Why is that? Does he feel entitled to any information about his grandfather, or does he not quite see her as a real person with real feelings?

Here's the reason Grandpa spent his life fighting monsters: It's a combination of bravery and honor at the core of his personality.

Jacob's identity as Abe's grandson isn't normally an issue, but it is with Emma because she was in a relationship with Abe. Jacob can't figure out if Emma likes him for who he is, or because he looks like his grandpa (but without the wrinkles).