If Frodo had thrown the Ring in right away

[Arwen Imladris]

Just look at Frodo. The Ring was destroyed but he wasn't cured, was he? Nope.

True, but he did feel a great sense of relief directly after the ring was destroyed, did he not? And there are many possible reasons why Frodo did not recover. For one, he completly isolated himself when he returned to the Shire. He didn't really try very hard to let other people help him bear his burden, instead he chose to hide his pain. Perhaps Gollum would have also recieved a sense of relief, a lightening of his burden. I think that if Gollum was prevented from jumping in after the ring, either by an inability to move because of his loss, or because of being physically held down by Frodo or Sam, he might have had a chance. We know that Hobbits are incredibly resiliant, and I think that even at this part of the story, Gollum still had some good in him. It was buried deep, but I think that if he was shown love and forgiveness, he might still be able to live a productive life. Remember, Gollum had never (to my knowledge) been stabbed by a deadly Morgul-blade.

After they lost he ring, both Bilbo and Frodo aged quicker. Isn't it likely that Gollum, after having lived on the ring for over 500 years, would age very fast and die within a very short time after the distruction of the ring?

True, but Bilbo, who had the ring the longer than Frodo and used it still lived to be quite old. He left it when he was 111, and he lived for another twenty years after that. If it took that long for Gollum's age to catch up to him, then he would still have a new shot at life.