Ill start with the second one because it might be a little easier to picture. Opposite to what many believe, liquid water is constantly turning into water vapour (the gas phase). Evaporation occurs at the ocean, seas, rivers, lakes, etc. causing vapour to rise from these bodies of water and condense as clouds in the atmosphere. In a closed vessel, equilibrium is allowed to occur and thus the evaporation occurs:
1) Dynamically, and
2) At the surface of the water (though this is also true when not in a closed vessel).
So in more detail; dynamic equilibrium in this situation would mean that as some water molecules vaporise into the air, some from the air will also condense simultaneously back into the liquid phase. Because the water is not boiling, this only happens at the surface of the water. If molecules of water were escaping from the bottom of the container, or from the middle of the body of water, you would have 'boiling' which also results in gaseous water molecules being produced (but it happens at a much faster rate, and its visibly noticeable - i.e. bubbles of gas in the water).
So if you were to write this as an equation:
H2O(l) <> H2O(l) + H2O(g)
Using Le Chatelier's Principle, if you put only liquid water into a closed vessel and allow equilibrium to take place, the reaction will shift to the left (products) to form gaseous water while some liquid water also remains. Once this has happened, the reactions DOES NOT stop! Dynamic equilibrium takes place meaning that some water molecules vaporise, while simultaneously some condense back into the liquid phase, but the concentrations of the reactants and products remain stable.
You can imagine how this would work similarly in the example with the ice. The only problem here would be that the equilibrium would shift to the left if the vessel were at room temperature.
H2O(s) + HEAT <> H2O(s) + H2O(l)
This reaction is endothermic so heat is absorbed to melt the ice into liquid water. If heat is removed from the equation (i.e. the vessel stays at or below 0 degrees C) then ice will remain in the vessel.
Hope that helps!