Before I get into your "3 nagging questions" (which looks remarkably closer to 15

) I will chime in to get on board to hear about these "nine full casters"... which sounds to me like, mmmmmayhaps, a bit much.
Three questions nagging at me right now...
1) How important is it to have the related classes get totally separate write ups if they follow much of the same rules and tables and share, say, 1/2ish of the fluff?
How "related" are we tawkin'? I mean, all classes -full, sub-, half-, whatever- deserve their own write-up. If nothing else their differences need explaining/exposing. It's just a matter of how much write-up is necessary. An "Illusionist" (even if it's its own full class, versus a "speciality school sub-class of a wizard") needs the explanation that it "functions in [all, many, some] ways as a wizard but [differences, alterations, things that are not in common]." As for "spellcasting" and preparation techniques nearly always falls under the "Magic" chapter or "Using your Magic" segment at the beginning of the Spell descriptions. Something about "How casting works in Combat" or some such. Usually there's at least some lip service paid to the difference between "Divine" and "Arcane" power sources....if you have nine full casters, I imagine that section is going to be somewhat longer but doesn't need full explication within Class descriptions.
How bad would it be to have a Wizard, Bloodline Wizard, Witch, and full-caster Bard all written up together as the "incanters" or "arcanists" who study spells, but then have them pick a school, bloodline powers, a patron and hexes, or songs res[ectively?
I don't know that it would be "bad" at all. It depends on how you set it up...and basically, your preferences of how those things work.
If you do that, is it bad if some of the spells are lower level for one variant than the other?
I wouldn't think so. My homebrew setting/game has several spells that are available to a number of different casters, who use magic [i.e. practice magic/cast spells] differently, with the same spell effect at different levels [tiers] for their class.
Again, this is a personal preference thing.
Introduced in D&D 3e or 3.5 I guess. Carried through PF1. Seems like D&D 5e did away with it. I, personally, see no problem to it. Par example: the Cleric can Cure [Light] Wounds as a 1st level spell. Obvs. MY setting/game Druids (and Bards, I think, and Witches) don't get it til a 2nd level spell. Characters with my game's "Ritualist" Theme (any caster who can use spells with the "ritual tag"), however, could perform Cure Wounds, too, as a first level spell (or higher depending on how many ranks in the theme they choose to accrue).
Would you still refer to them as four separate classes?
"Wizard, Bloodline Wizard, Witch, and [full-caster] Bard? Well, the Wizard, Witch, and Bard I certainly would expect to be three distinct classes. If all that's separating the first two is a bloodline, then that seems like the kind of thing to be handled by a background/origin/bloodline kind of decision point mechanic that is independent (and doesn't warrant) a separate "class" all its own.
2) In Pathfinder, Magus fills the "1/2 fighter 1/2 wizard slot".
Yes.
What is the name for a 1/2 fighter 1/2 witch?
In PF? That's still a Magus.
Should there be a 1/2 thief 1/2 caster?
A thief who uses magic or a caster who uses their magic to steal things? In my opinion, that's handled in D&D by what they call a Warlock. It's what the Arcane Trickster is suppsoed to be (and SHOULD have been). It could also be, in D&D, what is a Bard. Though I don't think any of them did it really quite right. Also, not necessarily a class concept, I feel, doesn't need its own base class...but an archetype that should be accomplishable via either a caster or thief base with appropriate feats/traits/themes/added options.
What is an all caster bard?
IN D&D 5e, that'd be a Bard. In PF too, I think. I'm not sure what you mean by "all caster." If all they do is use magic?...then in what way are they a Bard? "Wizard" would seem to be the proper answer.
3) Should all the caster types you find in the world actually be usefully playable as PCs?
Oh! No. Absolutely not.
I can imagine game worlds where there should be folks who enchant items, alchemists, people doing things with runes, and full-on healers.
As you should.
One option is to just have separate rules for them, but then you get complaints they do things no PC could ever do.
Which, since they are NOT "PCs" is perfectly fine and "in bounds."
Is it bad to have them have a class... like an old NPC class... that most PCs would never choose but I guess they could multi-class into?
Seems unnecessary.
Or does there need to be a variant like in PF where the alchemist can go all Jekyll and Hyde and throw explosives to make them combat useful but not fit the usual stereotype?
That is one way to go. Seems to have been quite successful for PF. But also sometimes leads to distinctions without [or with pointless] differences.
If there was a full-caster non-melee cleric, 1/2 caster cleric, and minimal caster paladin in the world at large, does the full-caster non-melee cleric need to have an option to make them like a blaster or super-buffer , or is it ok that there are just some character ideas that would stay at home?
That seems to be two separate questions. And, ultimately, again (sorry) a matter of personal taste and preference. I see no reason a full-casting cleric would have to be a "blaster" or "superbuffer" to go adventuring. It is equally valid to have full-casting clerics who DO go out into the world and those who choose/prefer to stay in their quiet cloisters meditating, praying, seeking/contemplating their deity's will, and/or serving their community and whatnot. The power granted to you as a cleric, afterall, is not for self-aggrandizement or personal pursuits (at least, usually, not from the Good or Lawful deities).
(Is it ok that the stay-at-home sage with a sponsor, friends, and budget knows a ton more about things in general than the bard who spends a huge chunk of their time fighting and singing too).
In my world/game? Absolutely. Again, your world, your preference.