4. Frog is many cultures closely connected with fertility. In the story about the Sleeping Beauty, for instance, a frog tells the queen she will after all the years of waiting give a birth to a girl. Frog (in several versions replaced by a snake) can also be interpreted as a symbol of maturity. A girl is old enough to get a husband but still young, inexperienced and immature (what is clearly represented with her need for a toy). She is willing to explore life, she talks to a frog and makes a promise, but she is not serious enough to keep her word and she is even disgusted with frog's appearance.
Marie-Louise Von Franz, famous Jungian scholar, discovered similar stories of humans being enchanted into frogs, reptiles or snakes in many different civilizations with very different look on human intimacy or animals, so it is probably better to explain her action with her state of mind (resistance to the change) than his appearance (and possible similarity with particular organs).
5. The above mentioned deal of the princess with a frog happens at the well (symbol of birth) in the woods. Although forest is doesn't play so important role as in many other famous fairy tales (think about Red Riding Hood, Beauty and the Beast, Snow White and others), this is again the place where the magical transformation of both main characters starts. Forest is popularly interpreted as a symbol of unconscious, a danger which is inevitable, if the character wants to find his/her true call.
6. Golden ball can be understood in different ways as well. Mythologists like to explain it as a sun (similar to the hood in Red Riding Hood), which is lost until it rises again - on completely new life for princess. It can also be explained as symbol of perfection (ball has a perfect round shape), maturity (it is made of gold after all) and carefree childhood which is lost forever when she drops it in the well.
7. King is another popular symbol which usually represents the authority. He is the one who orders the spoiled princess to keep her word. He is the one who says it doesn't matter to whom the promise is given, it is about her integrity to keep it. Please notice, there is no queen in the story. The plot doesn't need her. It is about the girl who promised to do something in exchange for something else and she needs a reminder which could be her inner voice (guilt) as well.