Fridge Logic: Futurians needed to bring 1992 era humans along in their mission to 1944... why exactly? They do everything themselves anyway and they already know from that book where they can find Godzillasaurus.
Mainly to let them witness the events that would come. No one hasn't seen Godzilla before he mutated except those soldiers.
Narm: The film has the most terrible acting from the English actors in the Heisei series, aside the rest of it. Especially with the American GIs in the World War II sequence. The acting was so bad, it fits the "So Bad, It's Good" territory. The English version, especially the English version has this in effect as well.
"Take that, you dinosaur!"
The English version contains one of the most bizarre statements ever. After seeing Godzilla kill King Ghidorah, some of the characters watching the battle notice Godzilla is starting to walk towards the city. In response, Prof. Mazaki shouts "Just look at that thing! It's not going to be friendly to us!". This suggests that he believes that if something is large in size, it's instantly evil. He's right about Godzilla not being friendly, but still, such a nonsensical statement is strange.
Most scenes with M-11, especially whenever he runs.
Special Effects Failure: Ironically, this film won for special effects. The SFX failure in question is when Emi Kano is using her jetpack, a rather unstably zoomed matte shot.
Tear Jerker: Shindo's final moments with Godzilla. After going out of his way to charge Godzilla with nuclear missiles (despite Godzilla going off-radar, unaware he still exists), he still sees him as his savior in World War II, giving a tearful salute. And when Godzilla rampages, he and Godzilla look at each other, with the big guy recogizing him. Shindo then accepts his fate, allowing his "suicide by Godzilla".