Reach is deffo not another planet, even in this, whatever verse this is. Its the home to the Spartens in all their forms, and its massive in agricultural etc, its basically Earth 2. Though none of this is conveyed in this show.
Also Kai, I have no idea what she's doing I guess she's replacing another arch character that went to train the Spartan 3s, but the Spartan that trained them had never even met MasterCHief at that point.
Yea, each of teh Majoinor armour is unique to the wearer, otherwise you have one suit does all, well its supposed to be uniquly wired for the wearers nervous system. But see as well they never even showed any of that, its..whatever. Even though they still use all the well known Chief look and suit for the propaganda posters etc.
Also, reach falling so fast, jesus, it was a war that went on for ages, and the USNC lost about half of its fleet alone above reach in a battle, of course, again, none of that is shown, and apparently people can just leave and fly on past in their ships.
A big problem now is how weak Spartens look, they seem to be cannon fodder, not that special at all really. They still dont have any clue how close the Sparten 2's where to each other, they are closer than family. Kai if true to a character would NEVER have left her team behind, she would have ripped the head off anyone telling her to do so otherwise, the idea she just gets on a ship and leave knowing what would happen to Reach etc is frigging abhorrent to what Spartens stand for.