Wouldn't B double sharp just be C? I guess they'd need to call it "B triple sharp", which is just getting silly.
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.
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.
Well-temperament still has different intervals depending on key. Even temperament just uses x^n/12 for note n and basis frequency x.