How does time travel work exactly in this movie?

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Typically a time travel paradox occurs when the past changes in a way that makes the time travel never happen. Killing your grandfather means you won't be born and thus can't travel back to kill your grandfather. It's exactly what you're describing with your two loops, but your point of time travel is in the future instead of the present. But the future isn't guaranteed and it isn't a paradox to change the future.

Hypothetically, let's say no time travel happened in this movie. The alien ship's advanced computer simulated the future, saw that David wanted to "go back in time" and then changed its mind about abducting him. But it left David with the simulated memories (we know it can interface with his mind, e.g. the star maps), and left the little alien which coincided with his fake memories.

I mean that COULD have happened, and it would be indistinguishable from what we're shown in the movie, and there's certainly no paradox.

So why would it be a paradox if instead of simulating the future with a computer, they somehow traveled to the future and then back to the present and the memories were "real" even though that particular future will never happen?

The point that he never travels back in time past where he started from is similar to the main concept of the awesome time travel movie "Primer." I think it's a critical requirement for avoiding paradoxes while retaining free will.

............what? max takes him back in the alien spaceship. pee-wee herman laughs, credits. what's not to understand? lol

"...Hey, at least now we know when the next train was due"

It could work as a parallel timeline like the branches of a tree. When he goes to the future an "alternative future" is set up, but when he goes back to the moment of his abduction (the start of the branch) he resumes the original timeline.
Also, fling in the ship through time could protect him from the effects of time travel like a cocoon.

"Knowledge is cheap at any price"

There is no paradox.

He went to an alternate reality where he was considered a missing child. At the end when he tells everybody "Sorry, I don't belong here" what he really means is he doesn't belong in that particular reality.

Max got him back to his proper Universe, and only he retained memories of that other parallel existence. Oh, and the puckmarin.

The same sort of paradoxes exist in Terminator. The difference is in all the Terminator movies, we are looking at the altered timelines, not the original timeline. Who was John Connor's father before he sent Reese back? John had to exist before he could send Reese back at all, and yet in the timeline shown in the movie Reese is John's father. That would be impossible in the original timeline.

Time travel depends highly on not changing your motives for time traveling. The motive for sending Reese had nothing to do with fathering John, so the reason for sending him remains and no paradox is created. If the motive for time travel disappears, then you never went back in time to begin with. The timeline is restored to its original state.

Example: Sarah is obsessed with preventing the war. If she succeeds, a terminator will never be sent back. This means she will never have knowledge of the war, which means she will never prevent it. The timeline is restored back to the beginning, where war is inevitable. Only now there is an infinite time loop destroying/preventing the future.

dude...it was just a film...