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.
Answer: Kate asked him to wait, and come in 2008, so that's what he did. Likely they could have found each other before then, but it might not have even occurred to either of them.
They started communicating after he died in 2006 (her time) on his way to see her, but 2004 his time, neither of them knew he had died by this time so there was no way they could have met, my question is how did he know to go and meet her the first time if they hadn't started communicating yet?
They had been communicating for 2 years at that point - his time. Hence why he was going to meet her and got killed in the process.
Question: Why would Alex put a letter in that mailbox? Did he know this lady was going to show up?
Answer: The mailbox is where you would send and receive mail.
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?
Answer: In her apartment in Chicago. Because Alex knew her address, he hid it in the floorboard for her to eventually find. He knew how important the book was to her. He didn't want to send it to her in the mail, as she originally requested.
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?
Answer: She mailed the latter at a later time, but asked him to call at the time she was still sitting at the table.
Question: The scene where Kate is sitting at the bar talking to the other female doctor and it gets quiet, Kate seems to realize something and then, "Oh my God" to herself. What is it that she realized?
Chosen answer: She remembers the guy at the party. He tried to get her attention. She dissed him in her misery. They talk about a book called Persuasion... She says it's her favorite, all about a couple who keep meeting in the wrong time and place and you never know til the end whether or not it will work out. They agree that it's a terrible premise... then they dance in the dark... and end up in this really sexy kiss that Morgan and Mona walk in on.
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.
Answer: When Kate and her boyfriend broke up and Kate was living at the apartment (2510 N Racine), the dog is there with her. It is assumed that the dog remained with her from 2006 onward.
Answer: The movie is not about them trying to find ways to meet. They are not trying that hard to meet and the time distance is a metaphor. I think the movie misses the tension that she is trepidatious about meeting and he is trying to respect that. He gets killed the first time for trying to force it! This shows when the stack of letters is piling away. She is worried he is a coward who will not be able to handle real romance. This is shown in his retreat to build mass produced houses and has something to do with his mom/dad. She's the doctor, he's the architect: she saves people for a living and is hung up about her own father's death. It's only when it's life or death that she gets over it and he also figures himself out and stops trying to fix everything.