1. Remember, 10 years prior to that episode, they didn't have such a complex and accurate checkout system. The library probably even has no idea who checked out the book. And even if they did, would it really be worth calling the police over?
2. He could, but the point is that Arthur isn't just looking for the ending. He wants to READ the ending. He wants to get all the excitement and stuff. Also, the Arthur series really didn't embrace technology much in that time...since they seemed to be trying to base the series in a world where all the modern day technology doesn't exist at the force it does today.
3. This isn't some new print book. It is old, most likely one published quite some time ago, possibly even 50 years ago. Therefore a lot of copies aren't still existent and in good condition. Keep in mind other things could factor in, such as that only a limited number of copies were printed, or something else weird.