The third movie is particularly convoluted not in itself, but in its position in the timeline, for it both follows and precedes the other two. Thus, it will help us to keep in mind the structure of the story based on the information established in the first two.
First, let us get it clear that the third movie presents us with three distinct main time lines. The first of these is the unaltered original history. It is unarguable that before Doc Brown could go back in time, history had to progress to the time from which he begins. Therefore we must have a history in which Clara Clayton and Buford "Mad Dog" Tannen and Shamus and Maggie McFly all lived their lives without any interference from Doctor Emmett L. Brown or Marty McFly. Of this timeline we know very little, but the little we know will be helpful. Once Doc Brown returns to January 1885, he creates an anomaly, a second main timeline, which is intrinsically altered by his presence in ways which are discoverable. The third main timeline diverges from the second in September 1885, when Marty McFly returns.
The only piece of the first main timeline which is important to us involves Clara Clayton. We know that in some timeline, Clara Clayton falls into Clayton Ravine. It is patently not the third timeline, because we see Doc rescue her. It is also apparently not the second timeline: had Clara fallen into the ravine in the second timeline, she would not have been able to erect a tombstone for Doc. Therefore she must have fallen in the ravine in the first main timeline, giving her name and the local story to that rift.
It will, I think, be accepted by all that at the beginning of the first movie, in the original timeline in which George McFly is hit by Lorraine's father's car, the first major timeline of the third part is the history. In fact, our first major timeline will have continued beyond that point quite a distance. As a general rule of thumb, any temporal anomaly caused by a trip into the past extends to the point in the future from which the time traveler began. However, in this story, because of the convoluted timelines of the second part of the story, the point in the future constituting the end of these anomalies is not so easy to divine. Consider for a moment: Doc left 1955 to return to 1885; Marty also left 1955 to return to 1885. However, both the Doc and the Marty who made these trips began their temporal travels in 1985. Furthermore, they had both been to the future at least as far as 2015. And this is the far end of our timeline. Ignoring, for the purpose of the story, the fact that the second movie crippled the timeline, we will assume (arguendo) that all of the trips made were possible. Thus we perceive that Doc and Marty would not have come to 1885 had they not been in 1955; they would not have come to 1955 had Biff not altered the timeline; Biff would not have altered the timeline had Doc not taken Marty to 2015 to help his kids. Doc would not have picked up Marty had he not already looked into the future somewhere beyond that point. Therefore, both of the temporal anomalies extend from 1885 to somewhere after 2015.
This means that the first major timeline brings us from the death of Clara Clayton to the original meeting of George and Lorraine McFly, on to the point when Marty escapes terrorists in the time machine, then follows the sawtooth snap created in the first movie by snapping back to 1955, and creating the timeline in which George decks Biff--the affluent timeline--and sawtooths until it stabilizes into an N-Jump termination. Time then progresses to after 2015, as Doc explores the future. All of the impossible mess of the second part becomes part of this timeline, up to the moment Doc is struck by lightning. At that instant, in that branch of the convoluted mass of temporal anomalies, the first main timeline ends, and is replaced by the second.
In the second main timeline, as we mentioned, Doc prevents the death of Clara Clayton. Although it is colorful to imagine that he rescues her dramatically as we see in the movie, it is far more likely that he meets her at the station, preventing her from driving the horses to town, being spooked by the snake and carried off the cliff. Neither of them were aware that he had saved her life. Doc had previously sent the letter to Marty, two days before Clara arrived. By that night, the two had discovered their common interests, and were enjoying the night at the town festival. But, in events similar to those seen in the film, Buford Tannen fires a bullet from a derringer into Emmett Brown when the elder scientist stands up for the lady. She undoubtedly remained close by his bedside, talking with him and comforting him, until he died two days later. Although Clara Clayton did not drive into the ravine, she did not marry, have children, or severely impact the timeline in any other way. By 1955, the changes made were minimal, consisting principally of a tombstone and a few newspaper articles and photos.
Now we reach one of the peculiarities of this anomaly. To accept this, the reader will have to perceive the deterministic nature of temporal anomalies, and the nature of determinism itself. Let us assume that, the world being a certain way, a person must make a choice between thing one and thing two. Let us assume that for any reason or for no reason the person chooses thing one. That choice having been made suggests that the person ultimately would not have chosen thing two. Thus, were we to erase all of the events leading up to that choice, and repeat them such that every event and circumstance which was relevant to the choice was the same, the person would again choose thing one, because the reason he chose thing one the first time would still have the same force this time: because this time is still the first time, even though we perceive it as repeated. No matter how many times the first time repeats, it will always be the first time. It is as if a stereo tape has been rewound, and the left channel re-recorded: the right channel will always play the same thing, no matter how many times the left channel is changed. As long as nothing which is changed is relevant to the present circumstances, the events will play out the same way.
Some will argue with this by suggesting that certain actions could be random. But if the choice between thing one and thing two was decided by a coin toss, the person would still conclude that the coin toss was the way to choose, and would reach into the same pocket at the same instant, grab the same coin, flip it the same way with the same force and angle at the same moment, catch it (or miss it) the same way, and get the same result. Even the wind which effects the way it flips would be the same. Thus, if nothing which is altered in the past matters to a specific chain of events in the present, those events will occur in the same way.
All of this is essential in order to understand that the second major timeline also leads to George McFly being hit by the car, Marty escaping to the past, the various travels of the second film, and Doc being hit by lightning. In the second major timeline, the only things which are different which matter to us is that there is a tombstone for Doc Brown not far from the cave in which the time machine is hidden, and that Clayton Ravine does not have that name.
This is a flaw that the movie overlooks. Marty should not know the name of Clayton Ravine, nor the story which accompanies it, because at the moment that Doc gets hit by lightning, Marty's entire history shifts to this second timeline. However, we will overlook this noticeable but small mistake; it is there for a plot purpose--the movie could not have any other way to tell us the name of the Ravine, unless it had done so before Doc was hit by lightning.
After Doc gets hit by lightning, Marty is in the second main timeline. The letter was not delivered in the first main timeline, since it was not yet sent (although the Western Union man will have won his bet several times, since our tracing of the temporal anomalies shows that this time comes once in the line in which George gets hit by the car, once in the line in which Marty interferes, once in the repeat of that line with the more affluent Marty interfering, once in the line in which Biff gives himself the book, and once in the line in which Marty takes it away--and each time in this second main timeline the letter is brought for delivery, but only in the last repeat is Marty there to receive it). The letter itself alters this timeline, as Doc now is instructing himself in understanding discoveries not yet made. Fortunately , we can accept the idea that Doc does not alter the timeline thereafter, choosing to make his time trip (either by intent or due to the availability of the plutonium) on the same day in 1985. But Marty chooses to end the second main timeline and begin the third by going back to rescue Doc, because he learns--and Doc in 1955 learns--that Doc is shot in 1885 by Buford Tannen.
Marty goes back and successfully saves Doc, without bringing him back to the future. Doc saves Clara in a much more dramatic way, and Clayton Ravine is renamed Eastwood Ravine, on the mistaken belief that Marty died there. And once again time progresses through all of the lines we've traced on these several pages. There are a few points which could change things drastically, especially at the end of these anomalies; we will look at these individually.
You might expect that the fact that Doc has sent himself the blueprints for the time machine will alter the future; you might expect that since Marty has told Doc about being hit by lightning, Doc will know to avoid it. However, you must remember that when 1955 comes around in the third main timeline, there is no reason for the entire sequence not to repeat. All of the events which we have traced through both previous movies repeat again. Then Doc and Marty receive the letter, and recover the time machine, and make the repairs. Unfortunately, this time they do not find Dr. Brown's tombstone. Dr. Brown built a new time machine, and took Clara with him into the time continuum. He was not shot by Buford Tannen, because Marty prevented that. Now Marty and Doc have no reason to investigate what happened to Doc Brown--and would discover nothing if they did--so Marty will do as Doc requested, heading back to the future to dismantle the time machine.
Unfortunately, on the day that Marty fails to return to the past, the timeline ends. Without Marty's return to the past, time is forced to return to the second main timeline, and Doc Brown is killed by Buford Tannen. All of history repeats its loops and swirls, until it becomes apparent that Marty now will go back to rescue Doc, restoring the third main line, after which he will not. The movie creates an infinity loop of the most awkward and convoluted shape, since it is filled with tags and jumps to different timelines; yet in simple form, time advances to just beyond 2015, snaps back to 1885, advances to 1955, and snaps back to 1885 to repeat the process. Once more we have destroyed time.
Is there any way out of this? It is clear that if someone goes back to change the past, they cannot both succeed and preserve their reason for doing so. It might be possible for Marty to have another reason to return to the past, and to save Doc, and thus to end in an N-Jump. However, we see no reason for him to do this, and it would take a true creative leap to find one here. Somehow, something has to happen between 1885 and 1955 which will spark Marty to return to protect Doc; but Marty no longer knows that Buford Tannen is going to shoot Doc "over a matter of eighty dollars", and so does not know to save him from that. No, for time to be saved, we must invent a reason for Marty to disobey Doc and return to 1885, and allow that having done so Marty will accidentally save Doc. It seems unlikely.
But perhaps, if there is a divine providence in these matters, there might be an intervention which would put the time line back on its track. There are records of the God of Abraham tampering with time--minor matter, mind you, but tampering no less--and He might do so again to save us all from the folly of our scientific tampering. On the other hand, I have said somewhere that time in the supernatural realm is multidimensional in ways which go far beyond our simple notions of forward, backward and sideways. Perhaps better solutions to the temporal paradox than the infinity loop, or to the impossible necessary future than the N-Jump, are found in this multi-dimensionality. And perhaps there is a Doctor Emmett Brown traveling in time who may read this and stop by to explain it to me.