Corrected entry: One crucial piece in Poirot's puzzle is the bathing cap that Christine reminds Linda to wear, so that she can't hear the noon-day gun, but there is no way such a thin rubber cap could be sufficiently sound-proof. (01:30:05)
Corrected entry: Poirot in his summing up of what took place in the murder of Arlena, makes many assumptions about exactly what took place on the beach, he said (in so many words) - Arlena went to the pedalo in order to leave the beach on it, but then ran in to cave instead. How could he possibly know this? There were no witnesses or evidence of this. All he knew was she must have gone into the cave as the gem stone was in there.
Correction: Have you never read or seen a whodunit before? Of course he is making assumptions. That's what detectives in these stories do. They come up with a scenario that explains all the fragments of evidence. To address your specific example, Poirot saw the cave when he was next to the boat. He would have reasoned that she was trying to hide from someone and must have seen the cave while she was at the boat. Why was she at the boat? She must have been trying to get away. Why didn't the movie take us step by step through this? Because the film makers didn't want to bore the audience by spending an hour on the revelation.
Corrected entry: Poirot finds out from insurance documents that the surviving husband of the murdered Yorkshire woman in the beginning of the movie, Alice Ruber, was called "Felix Ruber." Poirot had been told by hotel guest Patrick Redfern, a Latin teacher, that "Giuseppe Verdi" means "Joe Green" when translated into English. Poirot explains during the murder's solution that this triggered him to translate the two Latin adjectives "felix" (strictly: 'filix', not 'felix') and "ruber" into "fern" and "red", respectively, and thus "red fern." It is part of the evidence that Felix Ruber and Patrick Redfern are one and the same person, meaning that Patrick was involved not only in the island murder but also in the Yorkshire murder.
Correction: This hardly qualifies as trivia, as it's a major plot point, and is explicitly explained in the movie.
Correction: Actually it would stop the noise or at least muffle it enough that her splashing would cover the shot. Try it.