Explain why the water in a cup placed in a small ice chest (without a refrigeration mechanism) initially at –5 °C does not freeze.
There is nothing magic about absolute zero. In fact, it's not even a real, valid temperature - you can never cool anything down to absolute zero. You can reach temperatures arbitrarily close to absolute zero, but under no circumstances could you ever cool an ice cube (or any object) down to that temperature.
Supposing you could, however, your glass of water would still not freeze instantly. As soon as you dropped the ice in the water, heat would begin to move from the water into the ice. If the glass of water is small enough (or the ice is large enough), it might have enough capacity to absorb the heat from the glass of water until the whole thing freezes eventually. But remember, as the water is cooling down (and freezing around the ice cube), the ice cube is warming up. This process continues until the whole system reaches an equilibrium - either the ice has melted and the water has cooled, or the water has frozen and the ice has warmed. Which happens depends on how much ice and how much water. The process still takes time.