Ryan says that he's from Chicago, suggesting that the shop is located in that city. However, previous episodes contradict that, referring to "tropical storms" and Chicago as a place they called rather than lived in.
The first time the 1862 portal fades as soon as Rook enters it. The second time, it doesn't fade away until Rook arrives back in the present and blows the flame out on the lantern.
The photo that is taken of Ryan with Lee doesn't match the actual scene when the photographer takes the photo. The major difference is that Ryan is looking down in the photo, but when it's taken, he's looking up.
Jack says that all of the photos in the book are from Sharpsburg. However, in the next sentence Micki says that Rook tore out all of the book pages dealing with Sharpsburg. She subsequently finds a photo of Ryan at Sharpsburg.
Although Jack earlier told Howard that his name is "Atticus Rook", Howard doesn't act on it when he comes to face to face with the real Rook. In fact, Howard says that he didn't see Rook earlier.
Rook's plan has one flaw. He claims that the artifacts he brings back are "mint". However, since he brought them back via time travel, they wouldn't show any aging and any testing would show they were relatively newly-manufactured and not "antiques".