Elrond did not have a ring until sometime after the Last Alliance. Gil-Galad (name means, basically, "Starlight", though it has connotations of "Star-shine" as well) had the ring of Water, Vilya, until his death. So it's not likely that Elrond took the ring right off Gil-galad's still warm corpse. That's the kind of thing Men do...
Gandalf has Narya, the ring of fire, which was Cirdan's. Gandalf showed up about a thousand years after the Last Alliance
Some people are being unfair with the whole Turgon thing. If you'll remember, his friend Huor told him to leave because he knew that was the only hope for Middle Earth... which was true. Huor's son Tuor would marry Turgon's daughter Idril, and their son would be Earendil, and his sons were Elrond and Elros....
Any other queries I missed? I'm in database class, please distract me!