The Elder explains to Ladybug that in Japanese culture, ladybugs are seen as good luck because they take all the bad luck into themselves, rather than letting it harm those around them. That's basically what happened in the film. If Carver had taken the job as planned, he would have killed first and asked questions never, and the Big Bad's plan likely would have gone off without much of a hitch, and then Carver would have been executed when he delivered the case. Instead, every mildly heroic character except Tangerine survives the film (and of the twins, Tangerine is clearly the least heroic) while all the villains die.
Maria encourages Ladybug to reframe the way he looks at his luck; sure, a suicidal bellhop fell on his car, but Ladybug got him to the hospital and he survived. The bellhop was always going to jump—it was lucky that Ladybug was there. From that perspective, the fact that only one civilian bystander (the waitress) is harmed on the train starts seeming like more of his luck at play.
Look closely, and you'll see that the train crashes through the streets of the village at the end, improbably failing to do more than superficial damage to the buildings and the people inside them. Ladybug strikes again.
The Prince claims several times that she is Born Lucky (to contrast with Ladybug's bad luck). This is apparently something she inherited from her father, as the White Death has a habit of playing Russian Roulette with his targets, alternating between pulling the trigger on himself and on them. Obviously, he's never failed yet. Lucky indeed.
Or not. The White Death plays Russian Roulette in a very specific manner, implying that he's rigging the game, especially since the two times he's seen playing the game, the revolver didn't go off when aimed at his own head, but did when he was about to execute his victim. Also taking into account the tragedies that has befallen him, it seems that he's never been completely lucky, instead it seems that Fate was just taking a long time before killing him in order to make him suffer for his crimes.
And while she gets some lucky breaks, the Prince's manipulation and complex plans come to bite her in the ass multiple times, due to her targets seeing through her bullshit or people not doing what she had expected them to do (her plan was to trick her father into killing himself with an explosive-rigged gun when he disarmed her, since he traditionally kills assassins with their own weapon, but he doesn't do that right away due to his complete lack of respect for her).
Note that according to both the Elder and himself, the White Death fears the concept of fate, and instead does everything in his power to dominate the world around him. As explained under Fixing the Game on the main page, his displays of Russian Roulette are indeed rigged, and the encounters between everyone on the train, while they seem random, were mostly arranged by him well in advance, signifying that he's both a Control Freak and The Chessmaster. But of course, since no one can control everything, the White Death is still brought down by what he didn't expect.
The Son tells Lemon and Tangerine that his father doesn't need a reason to kill someone. He needs a reason not to. With all the problems the Son has caused over the years, once the White Death's wife dies he no longer has that reason.
The trains from "Thomas the Tank Engine" are used throughout the film by Lemon as semi-accurate metaphors for the various people encountered by him:
Tangerine is labeled as a "Gordon" and a "Thomas": Gordon is a rude, proud engine with a kind side; Tangerine is the meanest of the Twins, but still has a heart, as seen when he's devastated when Lemon seemingly dies and he spends his last moments trying to tell Ladybug that the Prince is a "Diesel" (aka, an evil POS). Lemon posthumously labels him as a "Thomas", who in the show is cheeky, but much nicer than Gordon, and the most iconic of the characters. Personality similarities aside, Tangerine wears a blue suit throughout the film, the same color scheme as the two aforementioned engines.
Before the latter's death, Lemon playfully labels the Son as a "Percy"; Percy is known for getting into various scraps in the show (getting caught in a flood, getting covered in jam and people's clothes, and nearly getting crushed by a boulder, just to name a few), similar like how the Son has caused numerous screw-ups in his own life that forced his mother and father to bail him out.
Diesel is used to symbolize an irredeemably evil person, such as the Prince, and like the Prince, Diesel is an arrogant jerk who has manipulated others for revenge or to get what he wants (ex: in one of his debut episodes, Diesel spread rumors about the other engines and lied that they came from Duck, in order to get him kicked out of the railyard).
After Lemon and Tangerine realize that the briefcase is missing, it's mentioned that Lemon is not wearing a bulletproof vest, but he's later saved by one. This doesn't make much sense, until you notice how disheveled they both are in the scene after Tangerine finds him knocked out. Lemon might not be the kind of person to bring a vest, but Tangerine is, and he gave Lemon his own vest the moment he realized they could both be in danger.
A common bit of logic for the movie is that the train takes all night to reach Kyoto, when in reality it should only take around a couple of hours for the trip. While it could be a bit of Artistic License, the actual movie is, credits and all, 2 hours and 6 minutes. In other words, it's a couple hours-long ride straight to Kyoto.
The Prince claims she is born lucky, and ladybugs are considered to bring good luck: Ladybug's accidental meddling saves the Prince twice. Not only that, but the Prince's luck runs out once Ladybug is made aware of her true nature, leading to her eventual death.
How did The Wolf survive the mass poisoning at his wedding? He spilled his wine on himself when Ladybug accidentally bumped into him and was cleaning it off when the poison killed everyone else. How does The Wolf die? He accidentally stabs himself in the heart while throwing a knife at Ladybug. Ladybug's luck seemed to have balanced itself out.
Fridge Horror
When the White Death is telling Ladybug about his plan at the end, after mentioning his wife's death Ladybug tries to offer his sincere condolences. One can only imagine what White Death thought at that moment, considering moments later we learn he believes he is speaking to the man who murdered her.
