Re the Cdim7 to C progression:
Note that C is enharmonically equivalent to B#, where the succeeding dim7 thirds are indeed D#-F#-A. So if you wanted to be pedantic, you could write B#dim7 instead of Cdim7. Then B# would "lead" to its enharmonic equivalent C (root), D# to E (third) and F# to G (fifth); you could even say A leads down to G. But however "correct" this would be in reflecting theory, virtually every fretted instrument player would simply chart this as Cdim7.

If I have doubts, I name a dim7 chord according to the nearest chord (usually a dom 7) that would be my second choice. The root is either identical or a semitone off, and the harmonic function is very similar.