I think the main purpose of Aragorn to even be in the story (other than to protect Frodo, same as everyone else in the fellowship) was to reclaim his family's throne in Gondor. So his being the son of Arathorn had a pretty big impact. Frodo's introduction, Frodo son of Drogo, I think was probably just his habit. He was more of an upper class hobbit, and when he sent a letter or did anything more 'formal' in the Shire, I bet he signed it Frodo son of Drogo. Sam, when he used the more formal greeting, I think it was him trying to prove (maybe to himself) that he was as good as all the other hobbits, even if he was 'just a gardener'. Legolas really didn't introduce himself all that much, and I agree that maybe he was downplaying the fact that he was an elf. Plus, he may have been trying to make a name for himself without his famous father. Just my oppinion though
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