This is a longshot, but my guess is:
Drumstick (musical drumstick, chicken drumstick, drumstick tree, ice cream treat drumstick)
There is a plant called a "Drumstick Tree" that is quite green.
The skin of a chicken drumstick, a musical drumstick, the ice cream treat, and the bark of the drumstick tree are all brown.
I cannot be eaten.
A drumstick for music is not edible. Many parts of the drumstick tree don't have culinary uses.
A chicken drumstick is definitely yummy though, as is the ice cream treat. Also, the fruit, seeds, and leaves of the drumstick tree have culinary uses.
Chicken drumsticks are served hot and so are the fruit and seeds of the drumstick tree.
Drumstick tree leaves are served cold on salads. The ice cream treat is definitely cold.
I'm hitting my front with my back.
You hit a drum ("front" of the word) with a stick ("back" of the word)!
Hint 2 & 3
Hint 2: The fruit, leaves, and seeds of the drumstick tree are refrigerated and the ice cream treat is frozen. A musical drumstick is used on stage. A chicken drumstick is often served on plate. Hint 3: There is one musical use for drumstick (drums), and three for edible (chicken, drumstick tree, ice cream treat).