Why can the letter c stand for two different digits — as in "ice" versus "cat"?

Because c has no sound of its own: soft c sounds like /s/ (0), hard c sounds like /k/ (7).

• ice → the c is soft, /s/ → 0.
• cat → the c is hard, /k/ → 71.

This is the "sound not spelling" rule in miniature. The same trap applies to g (soft "gem" = 6 vs hard "goose" = 7) and to s (hissy "sun" = 0 vs buzzy "daisy", where s sounds /z/, still = 0). Say the word aloud and let your ear pick the digit.