Here are my thoughts:
1. I'd definitely call it an S, based on what Ultra said (which alone settles the matter), but also because Gaia Dragoon never had a "second-half" counterpart for any of the series (no V2, GT, or MF, though it's not alone in the latter two).
2. This one is actually causing me headaches too, along with a bunch of similar things from RB's, in this case because Ariel 2 always comes with Star Attack, for example. However, for the most part, the weight disks seem to be random, yes. Go with what Ultra said here.
What I assume happens in this case, is that for a prize beyblade, the whole beyblade is placed in the booster, whereas for others, the assortment of parts are colour coded and thrown in randomly. But then there's things like poseidon/apollon/venus coming in the prize colours with the wrong bases. So honestly, no clue.
3. For the official/correct name for the part, Kai-V might be the person to ask, but Sub-AR is the accepted term. For SG Wing Base though, I'm not sure what that is called, I've always called it a sub-ring. In Terms of each SAR's name, all but one don't have a specific name and are called the same thing as the AR (only just realised only one has separate names for each). Zeus is the odd one out, with separate names for each part. No clue why.
Plastics naming is notoriously obscure, and only gets worse the deeper you look, especially when you reach Guardian Driger. It's AR, "Great Tiger", has the same SAR as Great Dragon (Gaia Dragoon) and SG Wing Base (Gabriel), but the main AR is a War Lion (Galeon) - so, it's given a new name, despite being made of recycled parts. Yeah.
4. Hidden Spirits are basically a separate side series, yes. They seem to have their own mythos/story, too.