Corrected entry: Kate asks Alex to rescue her copy of Jane Austen's 'Persuasion' from the train station bench. Alex tells Kate he will save it for her for when they meet. Then, in her apartment bedroom at 1620 Racine that she's sharing with boyfriend Morgan, Kate checks a loose floorboard to find that book hidden beneath; though the story makes clear that Alex has never been inside the apartment.
Corrected entry: Kate is in the future and has named the dog Jack and informs Alex of this in a letter, which he receives in a magic time-travel mailbox. Alex adopts this same dog in the past, before the dog has ever met Kate. Jack tests out the name "Jack" on the dog, and the dog reacts to this name. The dog should not know this name as it hasn't yet met Kate. This is backwards. Alex should have been the one to tell Kate of a name he gave to the dog, and she should have been the one to test it on the dog.
Correction: Jack the dog is magical, like the mailbox. Neither character knows where he came from. The dog is a sort of guardian. He runs to the past from the future. Re-watch the film, and this is clear. This interpretation is important for enjoying the film, if even more illogical than a magical Lake house without a magical dog.
Corrected entry: Jack the dog appears from nowhere early in the movie and is taken in by Alex. In the future, Kate, who has not yet met Alex, owns Jack and mentions her name in a letter sent to Alex. Since at that point in the film their paths had not crossed in either present or future, where did the dog's name come from? For that matter, where did the DOG come from?
Correction: I remember her saying " come on Jack, come on Jackie!" Jackie was said very quickly.
Other mistake: Every time somebody climbs to the attic in the "lake house", you see a small, dark, enclosed chamber with some abandoned stuff. However, as soon as you see the "lake house" from the outside, it is a single-level construction entirely made of glass, with no inside staircase or ladder, and three smaller glasshouses on the roof. There is not a single wooden panel to enclose the attic, nor any construction in the house that looks like it could be that attic.
Suggested correction: It's not a "dark, enclosed chamber"; it's a glass surround with part of the windows darkened, but others letting light in. The ladder is shown near the end of the movie (when he's putting the box of letters away) to fold up into a small space, and the "attic" is merely a small space, easily the height of the roof plus one of the glass bits you can see from outside the house.
After looking at many photos of the house, there is no space for the attic.
Correction: When I saw the scene in which Kate discovers the book under the floor, I immediately thought that Alex, at some point, went into the apartment building while it was still in the process of being built and furnished and hid the book there. He, as an architect and builder was able to pull it of and thus fulfill his promise of bringing her the book. He did it sometime after they broke up and before Kate moved in. I therefore believe this was not a plot hole but a clever idea that fits the movie perfectly.