Necron is a physical manifestation of death, sort of we think. Chaos is a demon lord.
The four crystals in I are elemental. In IX, they're all non-elemental (or rather Holy elemental).
The Four Fiends are there as fanservice, as IX deliberately homaged many previous franchise titles and took the series back to its European fantasy roots after VII and VIII's more modern / postmodern settings.
This actually has sketchy canon confirmation.
This is canonically true, but I tend to leave Tactics out of things since, despite being one of the most beloved, it is a side game.
The lore of VIII is pretty sketchy, or rather, more deliberately spelled out in an Ultimania, but that's because the main focus of VIII was Squall and Rinoa's romance. We don't need an in-depth history lesson of VIII's world to understand that.
In III, Hein (not Hyne) was a minor villain who uprooted an important tree (whose name escapes me at the moment) and used it as a mobile fortress. The Warriors of Light offed him, so he couldn't pass his power down to future generations of women.
Nothing falls from the moon in III. The Crystal Tower is just there, unexplained, like many things in the game.
The Lunatic Pandora was built around a Crystal Pillar Esthar excavated for the purpose of making the Pillar mobile, that Sorceress Adel might use it as a weapon. Nobody's quite sure when the Pillar fell from the moon, but it can be used to trigger calamitous Lunar Cries, so the Esthar government under Laguna sank it. (Then Seifer got it back during the game's events, and... yeah.)
True, but III's villain is named Hein, not Hyne. Semantics aside, Hein is a minor villain and was killed in the first half of III.
Gardens raising SeeD to fight Sorceresses was an idea a time-displaced Squall gave to Edea Kramer after the final boss in VIII, making a time loop.
Surprisingly, comparing the two maps between the games does show some similarity...
... but anyway. This is pure speculation. It's a possibility, but there are endless possibilities. As I've noted twice before, Hein in III was killed by the Warriors of Light about halfway through the game, and had no lasting effect on the storyline. If he were VIII's Hyne, I'd imagine he would have a greater impact on the story. He's just a flunky to make the Scholar class necessary for a fight. (Hein changes his elemental weakness often, and Scholars can identify it with ease. This is also why XIV Amon - who was based to a degree on Hein - can use every elemental spell in his fight.)
EDIT
-Hein and Hyne are spelled the same way in Japanese, so the mistake is understandable. Still, III Hein was a minor villain drunk on the power of darkness who was killed off almost immediately upon his introduction, while VIII Hyne is more or less the original god of the world (hence Sorceress' ability to use real magic instead of relying on Odine's para-magic draw system, since they have a small fragment of his essence).