Summer Wars/Headscratchers

Having dinner is a free action

 * Sakae's will says she wants the family to sit and eat a meal together as a family, which is a genuine Tear Jerker request. Except they all decide to do it right then, after Love Machine has taken control of a rogue satellite and has clearly announced his intentions to have it slam into a nuclear power plant, resulting in a catastrophic, Chernobyl-esque release of radioactive material. Apparently family togetherness is more important than saving millions of lives. Also, if they'd opted to eat the meal after dealing with the threat, then they would've had more than an hour to redirect or even save the satellite, as opposed to a ten minutes.
 * The worst things in this world are being hungry and being alone. They wanted to be full and together. Not the best of explanations, but they aren't a very technologically savvy family who just lost the most important member, and they had only recently started taking the situation seriously anyway. The main page pegged them as Too Dumb to Live after all.
 * Also, in the last moments, while trying to break the encryption on the satellite's guidance system, the same pattern came up: they break the encryption, you have a couple of seconds, then Love Machine puts up a new encryption. So rather than wait until they had the data they wanted to upload ready before finally breaking the encryption, then inserting it in the brief opening, they kept trying to brute force it, and very nearly getting the entire family killed in the process.
 * Kenji probably wasn't really thinking straight, since he had to focus so intently on solving each encryption (though to be fair to him, the Windows-style interface might've been such that he really couldn't do anything on the screen before pressing enter), but the dinner made sense. The scene only lasts a minute or so and they're gobbling down food as fast as they can while hashing out a new plan to fight Love Machine. The countdown started with nobody having a clue what to do next, so they were bound to need some time to figure out a strategy. They just did it over a quick meal.

Longest pitching run ever

 * Running parallel to the events of the movie is the baseball tournament Ryohei's team is participating in. Ryohei is the team's star pitcher. Instead, the movie shows Ryohei pitching four consecutive complete games, including one that goes into 15 extra innings. Considering that the tournament is said to last 10 days, not counting the championship game, the movie leads you to believe that Ryohei pitches a complete game 11 days in a row. Starting pitchers are given at least 3 days rest between starts, especially after heavy pitching days, such as a complete game, in order to avoid catastrophic arm injury, a very common occurrence with starting pitchers. Also, doesn't this team have any other pitchers?
 * You're thinking of US baseball. Japanese baseball has a different mindset - it's accepted to keep playing the same starting pitcher until he drops.

Koi Koi math

 * When Natsuki starts her game of Koi Koi with Love Machine, the bid for the opening round is exactly as many profiles as Natsuki has. Each round, the betting doubles, meaning each round Natsuki is going all in. However, when she loses a round, which happens before anyone from the outside joins in, she still has profiles left to gamble with.
 * She put her entire family up as her account, but the initial hand's rate was 1 point = 1 avatar. Koi Koi was in creasing the 1 mon = 1 avatar ratio. Second hand she started at 1 mon = 10 avatars. Third hand she set the bet at 1 mon = 100 avatars. Over the course of the hand koi koi calls had brought the total to equal 1 point equals 10000 avatars, when she lost, she must have lost by at least 30 points because she had more than 300000 avatars at that point, but she didn't have a number readily divisible by 10000 and so she still had some left.
 * The last bet confused me. Even forgetting why Love Machine felt the need to bet 10 million accounts rather than stall for a while, how the hell did Love Machine lose all of it's accounts in one move when it supposedly had 400 million ?
 * Goko + Aotan + Akatan + Ino-Shika-Cho is 28 mon, plus 1 mon per additional tanzaku (up to 3, 31), likely another 4 for tsuki-fuda (35), 6 more tane would yield 5 more mon (40). Kasu could yield some more (Up to 15, 55). 40 mon is possible, just incredibly unlikely. 40 mon * 10 million accounts per mon = 400 million.
 * Since Love Machine had lost almost every hand it played against Natsuki, it probably realized that it's vulnerable against her and needed to get out of the game fast. Love Machine didn't even want to keep playing after its first win, but the world rallied to Natsuki's aid and pushed her back into the game. Since it had to play one more hand now, LM gambled on being able to win a second time and so put up everything to make sure she can't keep the game going. It didn't work, but it made sense. As for the math, though, I'll leave that to someone who knows more about Hanafuda betting rules.
 * No, rushing the game doesn't make any sense, since Love machine would literally win by stalling. The only (and illogical) reason I can think of is if Love Machine (or in this case, the writer) wanted to give the heroes a fighting chance.
 * Love Machine wouldn't win by stalling. It hadn't set the satellite to hit their house yet, and it probably couldn't change the trajectory until it was free from the game (that might be another reason it made that bet - win or lose, it wanted to get back and set their house as the new target before the satellite hit the atmosphere). There's also the possibility that Love Machine was simply bored with the game, since it wants new information and Hanafuda had stopped being a novelty.

What was the military thinking?

 * Why the hell did the U.S. military decide to release Love Machine into OZ as a test? I mean, seriously, the thing is used by most world governments and businesses. Unless you plan on crippling the entire world (including YOURSELVES) during a war, that isn't a good testing ground!
 * They seriously underestimated what Love Machine could do and wanted the plausible deniability that anonymously releasing it into Oz would provide. They probably thought they had a killswitch to stop Love Machine once the test run was finished, and one of the first things Love Machine did was disable it.
 * The test was probably to see if Love Machine could get into OZ at all. The military wanted to see if Love Machine was clever enough to compromise OZ, presumably so they could use it to watch the world's business and government transactions from the inside. Love Machine's strategy at the beginning was admirably low-key and subtle, just as an intelligence agency would want. After it got in, however, it revealed why it was completely and utterly unsuited to clandestine work; it had a flair for the dramatic, a major power complex, and a vindictive streak a mile wide. Who would have expected that from a brand-new AI? Besides every sci-fi fan ever?
 * Was it ever stated that Love Machine was released intentionally? Maybe it just broke out on its own.
 * Yes, Wabisuki said the Pentagon used Oz as a test for Love Machine.

Love Machine's first avatar

 * Why did Love Machine take over Kenji's avatar in particular? Kenji didn't solve the equation that gave him access to OZ, there isn't any reason to use his account aside from providing a Red Herring.
 * Unless I'm remembering this wrong, in the discussion that revealed that Kenji got the equation wrong, it was stated that Love Machine took control of any account that responded to the equation, whether that person actually solved it or not.
 * But even then, it only manifested as Kenji's avatar, and Kenji was the first suspect, even though others had solved the equation. Plus, when Natsuki won back the stolen avatars, the only ones that remained were Kenji's and possibly the original account. Why not retain a more useful account, like a government official?
 * Kenji might have been the first person to respond, and so he was the first account Love Machine hijacked, which locked it in as the default avatar (there might also be the original account, but it's probably a classified military account that doesn't have an Oz avatar).
 * Its original account might be the whole 'Cheshire cat smile' thing rather than a separate existence as such. At the end it only has two accounts, and appears yet again as Kenji's corrupted avatar.
 * Kenji may have been the only OZ administrator to respond; that was his original summer job. Even if he was a low level administrator, the easiest way to compromise a security system is from the inside, so law enforcement would instantly be suspicious of Kenji if Love Machine used his avatar.