Question

Do you think the water on the outside of the cup comes directly from the water and ice on the inside of the cup?

Answer 1

The water on the outside of a cup can come from the water and ice on the inside of the cup through a process called condensation.
Condensation occurs when a vapor or a gas changes into a liquid state, which happen when the temperature of the air surrounding the cup is lower than the temperature of the air inside the cup. The water vapor will escape from the inside of the cup and then cool down and return to its liquid state on the outside of the cup.
Alternatively, it can also happen when the outside of the cup is already at high humidity, so the water droplets that we see on the outside of the cup is from the surrounding air not from the inside of the cup
It's also possible that the cup was already wet, and the water droplets on the outside of the cup are from that initial water

Answer 2

Personally, no.

Step-by-step explanation:

When a cup has a cold liquid with ice in it, the outside will begin to get really cold from the cold ice on the inside, and it will start to freeze and turn to ice as well. The warm room or temperature or your hands when holding it, will keep it cool just enough to make it cold, and damp.

Example:

Lets say I got a coke at wendy's or something, and when I got the drink, the liquid from the inside wouldn't be able to seep through the plastic. Even if it did, the plastic would be sticky and leave a sticky feeling on your hands, but you will notice that it does not.