Why didn't Gollum die when he didn't have the ring?

Okay so, it's been a few years since I've read the books or seen the movies but I was thinking about it and why didn't Gollum die when he didn't have the ring? And also while I'm on the subject why did Frodo get on that ship at the Gray Havens and go to Valinor? He wasn't an elf.. Neither was Bilbo.. I'm just really confused lol.