Spoiler Alert if you have never seen this movie, stop reading now. This answer is going to have multiple specific details pertaining to this movie.
Ok. Everyone gone that doesn't want to know this stuff? Good.
First... remember, whenever you step into a book or movie, you enter the universe of the writer. The writer creates the rules of that universe, and we as readers must decide whether we wish to accept those rules for the duration of our stay within the tale. This inevitably means that there will be things either our imagination or experience cannot readily accept. That said...
The movie does not explain away all the events as a simple dream sequence that Donnie wakes up to. If it were, he would wake up and proceed through life... with one or two small events in the conclusion of the story to make us go "Wait... was it really a dream?" In this case, the Jet Engine does indeed crash into the house, it does kill Donnie, and his family and friends (or potential friends, as in Gretchen) continue on through life as if he really did just get killed by a falling Jet Engine. So, the universe that the story takes place in, as explained by the events we are shown, really does contain all the pieces that enable what we saw to really happen.
That means for the purposes of a discussion on Donnie Darko, it is a given that what happens up to the point at the end where Donnie is laughing in bed could happen to a character in said universe. Although he has been diagnosed a paranoid schizophrenic by his psychologist, she has come to that conclusion from the information he gave to her... information he came by in the observation of events unfolding in his life... events that were not necessarily delusions.
It is possible that Donnie merely possessed the ability to see the future, and everything up to where he was laughing in bed before the Jet Engine killed him was simply a vision of the future. This would mean he did not have the ability to travel through time, and no actual time travel occurred. He saw events, and on realizing what would happen, he chose to stay in bed and die to avoid them happening. Now... I said it is possible... however, it is not probable.
The reasons it is not probable, are...
- because of the reactions we see everyone else have. Donnie laughs, Donnie dies, then we are shown the montage of people waking up with various stages or remorse and remembering. This tells us (the viewers) that it did happen. We are shown the page from Sparrow's book that describes this, and then we are shown it actually happening... the expressions of concern and the tears... down to Frank touching his eye in the remembrance of being shot there.
- because if it was simply a vision of the future, Donnie's vision would not have been "haunted" by Frank. Again, although it is possible that if he was seeing a vision of the future, he could have also been seeing a second series of future visions (aka Frank's appearances) within the main vision... it is not probable. He killed Frank. It is more likely that in a universe where one can (under extreme circumstances) time travel without technology, that Frank was also a time traveler, working to change events, just as Donnie chose to do at the end. Unfortunately, although this sounds good, there is a glaring flaw with that.
Now... there are several holes, even assuming the givens of this particular universe. For one... Frank appears to Donnie with his gunshot wound. That means Frank appears to Donnie when he was already dead. That's a gunshot wound to the head folks. So, how could dead Frank time travel, to tell Donnie about the future and actually assist Donnie in knowing what he needed to learn to change it? Also, if Donnie did travel back and remember it all, why wouldn't he get out of the room, and just avoid all the things that went wrong? Instead of laughing fatalistically, he could have gotten out of bed, written down what he remembered, and worked to make sure things didn't go that way. If he thought staying in bed and getting killed was a better way to handle it, he was already thinking one could change the past (IE staying in bed rather than surviving). If he thought one could change the past, then why not LIVE and change it?
If you step back and look at this from another point of view... assume Time Travel is possible through the power of the human mind. Assume that what we are shown is real in that universe. This story could instead be all about Frank. He runs over a girl, and gets shot. He travels back in time, he gives Donnie the message which inevitably forces the change that gets him un-shot (sorry about that).
I know what you are thinking... what does this have to do with your question... that and... if Frank never traveled back in time, he wouldn't have been shot, so there wouldn't have been a need to travel back in time. Yes. This is correct. However, whenever you are dealing with the concept of Time Travel, you must ALSO always deal with the possibility and consequences of a PARADOX. A paradox is a situation that is created by the probability of Time Travel, where it can and only would happen if Time Travel was possible... but otherwise is impossible. With Time Travel, and therefore with the Paradox, there is no "Chicken and Egg" syndrome. It is never a matter of which came first. The question of which came first is now a moot point, when you can just travel back before an event, to make something else happen first.
Example. Jill travels back in time and tells John to shoot the first person to walk through the door in front of him. The door opens, and John shoots and kills... Jill, since her former self was the first person to walk through the door. Your first instinct is to say that this scenario is impossible, because Jill wouldn't be alive to go back in time to tell John this, if John were to shoot her. However, your instinct is to say this because of the trouble we all have accepting the possibility of a Paradox. With a deeper understanding of both Time Travel and the Paradox, it is entirely possible for someone to go back in time and cause their own death before the point where they would go back in time. Not that someone would want to, but it could happen. It doesn't mean they cease to exist. It plays on the concept of divergent realities, and the concept that there are infinite universes created at every decision.
So... did Frank cause everything that happened to Donnie? Yes.
The Frank we are introduced to, even though we can't see what he looks like until the movie theater, is the Frank that was created by Donnie and the gun. It is the Frank of nanoseconds after being shot, assuming that the bullet kills him nanoseconds after that. Frank is the first time traveler in this story we are introduced to (but not the first time traveler in the story... the first time traveler in the story is Roberta Sparrow). Because of the way that the Paradox works, Frank pulled Donnie out of one time line (calling him out of bed), and led him down the path that resulted in his being shot. Why? To hopefully get Donnie to understand time travel as well, and get Donnie to decide NOT to shoot him. Remember, I'm not talking about the Frank we see on the floor after Donnie decides to stay in bed. This is the Frank from the time line where he is shot.
Donnie, with his mental illness, when faced with the same choice as Frank had (time travel or die), chooses to accept death instead. Roberta, when she is introduced to all of these things, she instead opts for a third choice. She becomes a new person overnight, writes it all down in a book, and becomes a science teacher. the Character of Roberta is there (among other reasons) to show us that Donnie could have opted to take a different path other than accept the Jet Engine to the face... but that's where his mental illness comes in.
So, yes. Frank time travels out of the time line of his own making to create the events that happen to Donnie and ultimately lead to his own death. The presence of him in the Rabbit costume WELL before Halloween night, as well as the picture Donnie draws of him that his mother sees, show that Frank is real, and Frank is traveling through time. Maybe not physically... maybe he can only be seen by Donnie because he doesn't have all his marbles.... but he is there.
EDIT almost forgot about the Jet Engine itself. The Jet Engine comes out of the divergent time line involving the events of the movie (and from his mother's plane), and is supposedly sent back in time through the same wormhole that Donnie uses to time travel with. Unfortunately, Donnie was miles away when he actually traveled back in time. The vortex might have been forming over his house, but that's not where he was. It was where he ended up... but the vortex and subsequent wormhole should have followed him... or he should have returned to it at that moment he decided to travel through time. The Jet Engine most certainly traveled through that wormhole... it came from the future to kill him in the past. IE... more proof of the actual Time Travel theory, as opposed to the Future Visions theory.
This again points to the paradox and says "See what you made me do?" If Donnie never got out of bed and was killed by the Jet Engine, how could he be killed by a Jet Engine in the first place when it came off a plane hitting the atmospheric turbulence created by the vortex that was there from Donnie corrupting the time line and NOT getting killed by the Jet Engine? Answer? Paradox!