A bunch of (undead, vampire) super-soldiers have to protect humanity (the white race) from hordes of invading aliens. The threat is everywhere and omnipresent, even though the actual baddies are underwhelming and you can readily slaughter tens of thousands of them. They're almost all uniformly evil, apparently just for the evulz. They're controlled by the all powerful yet unseen (((Darkness))) and the only thing protecting humanity is a giant white golfball. A white golf ball. Violence is the only way to interact with anything, and violence and death are goals unto themselves.
I keep trying to play this game and keep ending up frustrated by how empty, brutal, and fashy it is. Your characters, based on their actions, would be complete pyschopaths IRL, and the bad guy factions are all empty caricatures of evil evil baddies including Alien Invaders (Cabal), dirty illegal aliens who steal things (Fallen), Space Muslims (Scorn, Hive), and whatever the taken are (lazy writing so they can re-use assets).
Idk, I'm kind of done with it. Grinding for loot so you can grind for more loot is fun, but the story is so dumb and unselfconscious, and not even fun. Like with WH40k it's fascist and it's glorious because it revels in the excess and stupidity of fascism, it's all a huge fucking joke that's completely over the top. But Destiny plays it straight and it's so... dour and boring and unpleasant. This is the second time I've tried to play it and the second time i've given up in disgust.
No they're not. The Hive (fascist space shrimp in a giant intergalactic pyramid scam - I'm not joking here, their entire hierarchy is a pyramid scheme except instead of money their currency is literally abstract commodified violence, which is literally power that makes them magic, because paracausal bullshit) are vaguely allied to it, but neither them nor it are exactly happy with that. The Taken are basically just enslaved empty husks created by Oryx or later Savathun, the Hive royals who are every bit as incestuous and monstrous as IRL royals would be if they were magic fascist space shrimp that genocided galaxies. The Cabal are fashy imperialists with armies of canonically cloned soldiers that are "born" fully grown and thrown into battle immediately. The Fallen are aggrieved settler colonialists with their own manifest destiny shit going on. The Scorn are basically just strasserites. The Vex are sociopathically impassive and don't really have any stake in things except "fuck paracausality and its stupid bullshit; light, dark, idc I just wanna grill (on the surface of a star. it's a space grill)."
Meanwhile the Traveler's a fucking technocratic liberal that's just like "lmao hey everybody paracausality makes for some pretty cool tech huh?" handing out power without regards for how it's used or who hoards it (thus the "Golden Age" for humanity was basically a space-faring cyberpunk dystopia), then it fucks off once its fascist rivals showed up and murdered everyone, and it only stayed around on Earth because Clovis Bray (an evil bastard oligarch and one of the biggest villains of the background lore*) built a gun big enough to plausibly threaten it, forcing it to fight and nearly die. There's apparently some background lore that explains both the Traveler and The Darkness as basically competing gods, one trying to grow a wide variety of life for its own sake and the other trying to winnow it all down to one ultimate killing machine that destroys all other life, but really they both fucking suck because the Traveler hands out world shattering space magic without any sort of vetting or responsibility and then it just fucks off leaving everyone else to pay for it, and the Darkness is just omnicidal social darwinist fascist bullshit.
* unreleased current season lore spoilers about Clovis Bray
Now he's the senile robot that makes some of your guns, and neither him nor anyone else knows it.
There's nothing that conclusively proves this, we have no way to know for sure why the Traveler stayed. But everything else you cover is broadly correct. There is a lot of interesting lore in the game, but it's both hidden away most of the time and rarely used for the actual in-game storylines.