Does ice melt faster when submerged in water?
Assuming the water and air are the same temperature (and that the air is mostly stationary) then the ice will melt quicker in the water. This is due to water having a higher thermal conductivity and heat capacity. In fact, this is easy to test: place a cocktail stick through an ice cube and lay it on top of a glass filled right to the top with (room temperature) water. The submerged half will melt quicker than that on top.

Regarding the second question: A warm object will cool quicker in icy water than in just ice for the same reason above but also because of increased surface area contact. However, icy water will be warmer than just ice. So the item will potentially be able to achieve a lower temperature in just ice (slow and steady wins the race).