To your question "Does the original version of the universe still exist where Marty simply disappears", I would say the answer is "no". I think I'm not really disagreeing with Axelrod's answer since that answer is focused on whether Marty could recreate his original timeline, not on whether it continues to exist in parallel with the original, and Axelrod notes that timelines "cease to exist when overwritten". But I thought it might be worth adding an additional answer which expands on the idea that BTTF features a single rewritable timeline, rather than each trip to the past creating a new timeline which exists in parallel with the timeline the traveler had just come from.
The key is to remember the part of BTTF II where old Biff in 2015 steals the DeLorean, goes back to 1955 to give his younger self the almanac, and returns to 2015 afterwards, all while Marty and Doc and Jennifer are wandering around in 2015. After old Biff gives his younger self the almanac and sets the DeLorean to take him back to 2015, he should end up in the future of the world where his younger self gets rich using the almanac--the future of the hellish "1985A" we saw in the middle of the movie. In a parallel timeline model, Marty and Doc and Jennifer would remain in the original 2015, and simply find that the DeLorean was missing. Instead, BTTF writers Bob Gale and Robert Zemeckis explain in the Official BTTF FAQ that 2015 transformed around them:
1.9: When Doc and Marty are in 1955-A, Doc says they can't return to the future to stop Biff from stealing the DeLorean, because it would be the wrong future. But if that's true, how did Old Biff manage to get back to the same future that he left? Shouldn't he have come back to a different future?
A: As should be clear from the answer to the previous question, we believe Old Biff DID indeed return to a different future — a "2015-A," which would have transformed around Marty, Doc, Jennifer and Einstein (just as Doc explains how 1985-A would change into 1985 and instantly transform around Jennifer and Einstein). This would happen AFTER Old Biff returned with the DeLorean. For this reason, we made sure that Doc had caught Jennifer and exited the McFly Townhouse before Old Biff returned. Thus, by the time Marty and Doc are carrying Jennifer back to the DeLorean, there COULD be other residents in that townhouse — or perhaps the McFlys still live there. It is just as believable that the physicality of the neighborhood did NOT change as it is to believe that it did — so we didn't change it. We decided not to make anything of this idea because this is one of those difficult time travel concepts that general audiences have a real hard time understanding. (Try explaining this stuff to your mother and you'll see what we mean.) A detailed explanation of it would have slowed down the story, and most of the audience doesn't ever think about it. That's why we made certain things ambiguous and left various things open for interpretation in hopes that the possibility of at least one or two explanations would be better than a "definitive" explanation that you could find holes in. Let's face it, time travel is fantasy, so there's no way to "prove" anything. As filmmakers, we try to create a set of rule for our stories and stick by them, and stay consistent within the little "universe" that we've created.
And as they note, the same thing applies to Jennifer, since they left her on the porch in 1985A before they realized the timeline had been altered, then once they realized what had happened they decided to go back to 1955 (followed by the accidental trip to 1885), and when they returned to the new "fixed" 1985 Jennifer was still on that porch. Doc seems to know that time travel works this way, since when they're about to go back to 1955 he has this exchange with Marty:
Marty: Doc, what about Jennifer, what about Einstein? We can't just leave them here!
Doc: Don't worry Marty, assuming we succeed in our mission this alternative 1985 will be changed back to the real 1985, instantaneously transforming around Jennifer and Einie. Jennifer and Einie will be fine, and they will have absolutely no memory of this horrible place!
Also note something similar would have happened near the end of BTTF II when Marty is in 1955 and sees Doc in the DeLorean getting struck by lightning, then is almost immediately greeted by the Western Union guy with the message from Doc in 1885. In this case, there wouldn't have been a Western Union guy driving towards Marty with a letter for him when Doc was still hovering in the sky, but then when the lightning hits and Doc goes back to 1885, the "ripple effect" would have caused the world around Marty to change, including the Western Union guy materializing on the road near him (along with Doc's gravestone materializing in the graveyard and the older DeLorean materializing in the mine, as seen at the start of part III, and probably the Clayton Ravine transforming into the Shonash Ravine as discussed in the answer to question 1.18 from the official FAQ).
So, all of this seems pretty clearly incompatible with the branching-parallel-timelines model, and seems to indicate a single changeable timeline. To really make sense of this I think you have to imagine a second time dimension to keep track of "changes" to the timeline, or "meta-time" as it's termed on this page about time travel theories. Then you could have timelines that are "earlier" in meta-time and timelines that are "later" in meta-time. And when time travelers like Marty and Doc get to experience different timelines, we can assume that timelines they experience later according to their personal subjective time (which is also the order that movie viewers following their adventures see) are also later in meta-time.
Incidentally, the chart Axelrod posted, which originally comes from this deviantart page, can be understood as a graph which charts ordinary time on the horizontal axis (increasing from left to right) and meta-time on the vertical axis (increasing from top to bottom), with lightning bolts representing cases of the timeline transforming around Marty without him using a time machine (once in 2015 and the other in 1955 with the Western Union guy). The diagram Doc draws on a blackboard in this scene from part II could be interpreted in the same way, nothing that he says in that scene indicates the two timelines are coexisting in parallel, although a lot of viewers have interpreted it that way. This meta-time stuff is a little convoluted but doesn't lead to any obvious paradoxes, and it hangs together logically well enough that someone was able to design a simulation game about temporal war using this type of model (see here and here for an explanation of how it works).
One last point about meta-time in BTTF is that perhaps we shouldn't think of the timeline at each point in meta-time extending infinitely into the future. Instead, you could imagine a little arrow on the timeline marked "The Present" at each point in meta-time, and perhaps imagine that when time travel isn't happening, this arrow continuously moves forward along the timeline as meta-time passes (at a rate of one second per meta-second, say). That way, the future could still be in flux even if there was a definite past history at each point in meta-time. This would allow Doc to be correct when he tells Marty and Jennifer at the end of part III that "your future hasn't been written yet. No one's has." It would also help make sense of Gale and Zemeckis' answer to question 1.13 in the FAQ, where they say that although the future isn't written, jumping to the future takes you to the "most likely future of the moment you left", a "logical extrapolation" of that moment.
Thinking about what happens to "The Present" when someone makes a time jump, I would imagine the little arrow jumps along with them, and then starts moving forward along the timeline again from the point it jumped to. If the jump was to the past, it will be continuously over-writing that section of the timeline as it moves, like a tape that's been rewound to a certain point and then recorded over from that point on (do kids still know how tapes worked?), though perhaps the memories of time travelers will be immune to being rewritten at certain key points on the timeline (see my answer here for my ideas about how memory-preservation could work in BTTF).