The biggest difference from one piece of ice to another is it's structure. Ice is a crystal. A crystal is just a fancy name for when the atoms and molecules are formed into an organized shape. A good example is a brick building. The bricks are laid so that they are in a pattern.
Every crystal has many defects in it.
When you let the ice take a long time to freeze, then it is shaped better. As for the last part of your question, anything warm will melt ice. The warmer, the better. Another way to melt ice is to cover it in salt. Regular water freezes at 32 Fahrenheit, but salt water freezes at a much lower temperature.