On the subject of "what if" conerning Gollum's death. Personally, I don't think it would've made that much of a difference in BEGINNING the doom of the One Ring. Think about it...what would happen whenever Frodo put on the ring? It would draw the attention of not only Sauron, but the Nazgul as well. Now do you think it was the realization that the ring might be in The Shire that made it so both he and the ringwraiths could just suddenly feel the power of the ring when used? Perhaps, but seems unlikely...
Now assuming Gollum hadn't been able to reveal Baggins/Shire, after Frodo was given the ring, don't you think eventually he'd want to try it on?? At which point, he'd turn invisible and enter the "netherworld," which Sauron could gaze into. I don't buy the idea that merely the revealing of the Shire made it so Sauron could tap into it when used, when in fact I believe he had actually just gathered enough strength at this point to be able to feel the ring when it was used; Gollum's admission was just a faster way to figure it out. On top of that, I also believe Frodo would look for his own answers, which I'm sure Gandalf would hear about at some point, in which the normal chain of events from FOTR would begin...more or less.
Now once we get into TTT is where it's trickier. I've stated before that even if Frodo/Sam hadn't come across Gollum, I'm sure they would've found some way to eventually get to Mordor. Admittedly I can't think of how, but....I'm sure if Gollum had never existed they'd still be able to find a way. Point being, because this is a good guy story, they'd make it to Mordor eventually.
So at this point Gollum's death by Bilbo doesn't seem like it would impact the journey to that great of an extent. However it is at the destruction of the Ring that it would get interesting. Would Sam wrestle with him? Bite his finger off? Throw him in? Just let him walk out? Hard to say....though I'm pretty sure Sam's role in the adventure would be MUCH larger without Gollum. And in fact he just might be the one to throw Frodo to the ground, tear the ring off his finger and chuck it into some lava, or in a more....subtle Tolkien fashion try and grab Frodo who, when pulling away, would trip and consequently be semi-pushed by Sam over the edge. Thus destroying the ring, but still in a heart wrenching way. Hm...