Pilates focuses more on strengthening than on relaxation. The exercise is more vigorous, and intensely abdominal. They are really complementary modalities. If you have done Yoga, you will find yourself enjoying Pilates, and vice versa.
Pilates is a great complement to your gym workout, as it will lengthen muscle groups, and build core strength so you are less likely to injure yourself. The springs and pulleys that are part of the Pilates apparatus provide support to provide a platform for a deep workout, without using a lot of pressure. The resistance is pretty light and you will find muscles you never thought you had.
Generally accepted medical standards dictate the the child must be wearing an adult shoe size, and have reached at least the age of 12 or 13. Much strength training must also have taken place for at least two years prior to attempting pointe work.
Generally, no. Tutus are for performance. Sometimes an advanced professional dance will wear a tutu in class when working with a partner so that they are both used to the feel of it when working on partnered turns and lifts.