The Terminator mission would have failed even without Kyle Reese being sent back

"The story itself is a paradox "

The story itself is not a paradox. If you want to call it one, you can say it utilizes the 'Bootstrap Paradox', but in my opinion, the 'Bootstrap Paradox', despite its name, is not REALLY a paradox, as it's pretty logical and everything CAN work out within it perfectly (if done right, and this movie does it 99% right).

I mean, there's nothing inherently paradoxical about it, and it's NOT like trying to lift yourself up by pulling your bootstraps at all.

Another movie, where Bootstrap Paradox is done flawlessly, is '12 Monkeys' (or is it 'Twelve Monkeys'?).

From 2029 people's perspective, Reese was always in 1984, and always stopped the terminator, even if he still waits 10 years before he goes through. There's no rush when we're time-traveling, because we control the point of arrival, which is the only thing that matters. Even if Reese hasn't YET been sent through, he has ALREADY arrived, because he WILL be sent later.

I hope you understand it now. It's not linear time, it's not "Reese will ONLY arrive in 1984 AFTER he has been sent", because he's not sent linearly to just some location, but he's being sent back in time.

The only way your thinking would make sense if Reese would NEVER be sent, but that's obviously not the case. Delays in the future won't matter, when the 1984 arrival point is the same.

"You are not thinking fourth-dimensionally."

Your BttF series line is dismissed.

"Also, it's an IT, not a 'he'."

First, even if the Terminator were strictly a machine, machines are commonly referred to as "he" or "she"; cars in particular are commonly referred to as "she". Second, in this case, we have a machine/organic combination who can pass for a human male while interacting with people and whose organic parts include male reproductive parts, and it follows that the cells which make up his organic parts have male chromosomes, so there's even a scientific basis for referring to him as "he". Third, characters in The Terminator (and its sequels) have referred to him (and other Terminators) as "he", even characters who know what he is exactly. For example:

Kyle Reese: "You still don't get it, do you? He'll find her! That's what he does! That's all he does! You can't stop him..."

Reese has also referred to him as "it", which is fine too.

"It has also NOT been there for 45 years, because it was DESTROYED, so it ceased to be a terminator at that point, it existed in 1984 for a very brief time."

Regardless of what happened to him in 1984, he's been there for 45 years. A broken-down car is still called a car, for example. Also, we don't know what happened to him in 1984 before Reese went back there, we only know that he couldn't have succeeded in killing Sarah Connor or doing anything else that substantially altered 2029, because nothing in 2029 changed when the Terminator went through the time machine.

"From the perspective of people in 2029, Reese ALSO has been in the past"

Only after he goes through the time machine. The point is, there was obviously no need for him to do so.

"because it doesn't matter when you SEND someone to the past, it only matters when they ARRIVE. Since Reese was 'destined' to be sent to 1984, Reese appeared in 1984 the same day the terminator did, regardless of how long a time passed between SENDING the two characters back in time."

Those things are irrelevant to the point that there was clearly no need for Reese to go back in time. The same thing applies to your next two paragraphs.

"Otherwise, who gets to the past first, would win, and change the future, and even if Reese is 0.001 seconds later, Reese would never arrive or be sent, and Skynet would win, etc."

Yes, whoever goes to the past first would win, assuming they can successfully complete their mission. By going to the past first, the Terminator had 45 years to complete his mission. It doesn't matter how quickly Reese may have planned to follow him, because if the Terminator had been successful during all those years he had to do it, adult John Connor would have been erased from history thus 2029 would be drastically different immediately after the Terminator went through the time machine. "Immediately" doesn't leave 0.001 seconds to react; it doesn't leave any time at all.

Consider this more extreme example to illustrate the point: suppose the Terminator's mission was to release some hypothetical super weapon in 1984 that would blast the Earth into quadrillions of little pieces. Regardless of whether he achieved success in 1984, 1989, 1998, or any other time before present day 2029, there would be no time for anyone in present day 2029 to follow him back in time to try to stop him, because the Earth was blown apart prior to 2029. And if someone in 2029 is able to follow him back, that inherently means he was never successful anyway, so following him back is pointless.

Your next paragraph and your post below where you replied to yourself are irrelevant to the point for the same reason as your other three irrelevant paragraphs.

Time travel movies ALWAYS have flaws in the flow of their logic. It comes down to what you will accept and what you won't.

This movie ended with that Twilight Zone-ish bit where Sarah looks at the photograph. It's the same one that we saw earlier in the movie. It's the only piece of evidence that suggests the future is set to repeat itself. You and the OP are just debating the same flaws in logic that have been made since this movie came out.

Cameron abandons this model in the second movie when we learn that 'there is no fate but what we make'. The new model informs us that the future WAS changed but not enough to prevent the rise of Skynet. It's a parallel timelines model. Each trip to the past starts a new timeline because of the cascading effects (aka Butterfly Effect) of changes to the past.

This model and movie also has its flaws with time travel. Everyone ignores them though for the sake of a good action flick. I won't bother with the flaws... lots of essays, blogs and vids go over that ground.

Looper was the best time travel movie that I know of that had a cohesive model... but it still had to make compromises for the sake of the story. It's the same one used in Frequency and Time Cop.

There IS a way to theoretically travel into the past and avoid paradoxes and such. If you're interested look up the concept of a light cone. It still wouldn't allow for time travel movies like this or any other though.

"REESE WAS ALWAYS THE FATHER there was no other original Father"

That doesn't even hypothetically make sense. And there could easily be an original father who wasn't Reese. It would just mean that the events that led to the original, different John Connor would have played out differently the first time around. It could have gone like this:

Original timeline: Sarah Connor gets knocked up by some random guy who isn't Kyle Reese, and there is no Terminator trying to kill her. She names her son John and he joins the military when he's 18 and makes a career out of it. He ends up being the leader of the resistance against the AI machines and happens to choose Kyle Reese to send back in time when the machines send a Terminator back to kill his mother in 1984. There is no gas station picture yet.

Second time around: Instead of Random Guy, Kyle Reese ends up knocking up Sarah Connor and the events play out similar to the movie. The picture is taken for the first time.

Third time around: This is the first time where the events could have been identical to the movie's events. And very similar loops could have happened many times after this, but eventually things change substantially as we find out in the sequels.

Doc invented the time machine to travel forward in time.