Why does my breath cloud disappear quickly on some cold days but hang around on others?

It depends on humidity. On dry, cold days your breath evaporates almost immediately because the air is thirsty for moisture. On humid, cold days the air is already saturated, so your breath lingers in dense, persistent clouds.

Can seeing my breath tell me if it's cold enough for ice to form?

Yes, to some extent. If you're seeing thick breath clouds that linger and it feels genuinely cold (not just fresh), you're likely in the 5-7°C range or below—not far from freezing. If there's moisture around from rain or damp paths, that's when you should start thinking about ice forming, especially overnight or on shaded sections.

Does visible breath mean the same thing at different altitudes?

Not exactly. The condensation point changes with air pressure, so visible breath at sea level isn't quite the same as at 1,000 metres. It's still a useful indicator, just not a precision instrument across different elevations.