Question: Why would Alex put a letter in that mailbox? Did he know this lady was going to show up?
Question: Why did Alex have to wait until Valentine's Day 2008 to meet Kate? Even assuming for the sake of argument that the "time-traveling" letters stopped after Alex avoided the bus accident on "his" Valentine's Day 2006, Alex certainly knew how to find Kate at that time - and vice versa. The two-year gap where Kate had not known about Alex had closed by that time, and each knew about the other.
Question: Where is Kate living when she finds the book under the floor board? If it not the Lake House how did it get there?
Question: How is it that they have to be at the mailbox to receive their letters, but that one night she writes to him to call her right then and there as if they are reading each other's minds as they write on paper?
Question: What happened to Jack the dog? Last she was seen was when Jack showed up at Kate's boyfriend's work. The boyfriend said "I know you." and looked back at Alex walking away. I assume that's how Kate got the dog. But Jack isn't really seen in Kate's time much, after the chess game.