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.
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.
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.