There's no black note between B and C, so B sharp would be the enharmonic of C. B double sharp would be the enharmonic of c#

There's no sharp/flat note between B and C - "B sharp" would be C.

Indeed -- curse you, irregular standard note structure!

mnemonic:

C D E F G A B [C]

is

do re mi fa sol la si [do]

mi (E) and si (B) end with "i" and are the two ones that are half a step away from the next ones.

Only in "fixed do" version though, because we can't make music notation too easy or consistent...

G is just 9-fold-sharp B, really

ultra-pedantically: that's true with the "well tempered" tuning that is used almost universally. But this is really just an approximation, and one that our ears have grown used to.

Mathematically speaking, B# <> C. On a piano or other instrument that quantizes tones, we have little choice but to go with the approximation. But for violinists and others that can choose from a spectrum of tones, it's possible to hit those tones exactly. Of course, when playing with others, the violin still needs to quantize just to be in tune with the others.

Even more pedantically, modern stuff uses equal-temperament, not well-temperament.

Well-temperament still has different intervals depending on key. Even temperament just uses x^n/12 for note n and basis frequency x.

If anyone wants to do down that rabbit hole for 15min, there's Adam Neely's recent video on the topic: https://youtube.com/watch?v=SZftrA-aCa4