Interpreting the C-Index
For the first Q you already give a counterexample:

It is biased to k, it prefers k = N. And it will also overrate N-1, N-2, ... So it is not agnostic to k.

If k-means would always find the best C index, then the C index would just be redundant to SSQ, which is much cheaper to compute... But you probably have just been looking at way too simple toy datasets. Use real data.