OP - you have got yourself a difficult challenge because you say that you have never seen the theatrical cut of Donnie Darko. This means you exist in a Tangent Universe in which a jet engine fell on the script and killed much of its charm. To get where others are coming from you will have to try and envisage an alternate you who is unaware of a lot of the director's intent not just going in but throughout, a film with a much more surreal film.
A lot of the charm of Donnie Darko is the surrealness, the disjointed nature juxtaposed with the underlying sense there is a meaning there just out of reach. Nearly all of your opening post is trying to analyse the possible ways it could all work and pick apart what is consistent and what is not, evidence for this or that. When I watched the original movie, my response was much more an emotional one. Regardless of any time travel or alternate timelines my primary take away from the film was an emotional resolution in which Donnie reconciles with his fate and we see him going to bed smiling because in his mind he finally has a resolution to his strife and confusion and it is the knowledge that by doing so, he is saving everyone else.
And my sense of Frank was not trying to pick if / when he was a manipulated live/dead, but a feeling that there were other planes of existence outside of time, that the boy who died had moved to one of them, was attempting to save the world via Donnie. The moment we see the real / living Frank working on his bunny costume and heading off to the party is a key moment for the audience in the theatrical cut because it's one more piece of the puzzle we can slot in place, it explains why Frank is a bunny, it reinforces that there's both tragedy and higher meaning. Well, honestly the audience can take their own meaning from that but this is kind of the point.
I came out of the theatrical cut quite delighted by the film and recalling great moments like Donnie's meltdown about the lifeline diagram and his parents in the principle's office, the weirdness of Frank in the bathroom mirror and touching the eye, the smile on his face as he lies down to die, and the little girls saying:
"I heard that the bathroom was flooded with faeces."
"What's faeces?"
"Baby mice."
"Awwwwww"
You have to try and imagine the added mystery and more emotional version of the film because you never had the absence of the things that made it so. It's easier for me to see it in your terms - a puzzle movie arranging a multitude of pieces - than I think it is for you to see it in mine - a surreal, dark and uplifting mystery hinting at larger meanings.
Unfortunately that's most of what you have to say. I think you wanted more of a discussion about what made sense, thoughts on your tapestry of hypotheses. But to me it was more an emotional movie than a paradox movie and when I showed someone the movie I'd enjoyed and naively picked out the Special Edition because I thought it would be a treat for both of us, I found myself increasingly disappointed. The man behind the curtain was not a wizard after all.
Donnie Darko had a lot of things that made it good. Jena Malone has always been a very engaging actress. Even as a child playing the young version of Jodie Foster's character in Contact you can see how talented she is. In a very understated way. She also has a penchant for doing weird and off the wall movies - I last saw her in The Neon Demon which oddly has some of the same wont-explain-itself quality of Donnie Darko. Albeit an utterly subject matter and quite different tone. Jake and Maggie Gyllenhaal are good as the siblings. There's real comedy value in juxtaposing Donnie's frustrations and the school environment and lessons. Seeing him angrily taking apart some of the woo-woo nonsense they're being taught was as cathartic for some people back then as seeing a kid take apart woke or diversity nonsense in a film would be today. And in general, the film just had a wonderful atmosphere and pacing that sailed between character comedy and tragedy and gave you a sense there was a meaning behind events and higher planes of understanding. The emotional conclusion of the film is a sense that things are now right with the world, even if in a sad way.
It's a good movie but for me at least, was more of an emotional one giving me a feeling of mystery, sadness, bittersweet satisfaction. The strangeness of the unknown was a tool to bring all these feelings about and by adding so much in the director's cut it both diminished that tool and also created a competing area of focus for it. And IIRC, the director's cut goes a bit more into trying to say Donnie is a superhero of some kind referencing "he will have increased strength" or something. In the original the axe in the school mascot's head is just this weird background detail.
Thank you for coming to my Sped Talk.