olive oil
In my experience olive oil has too strong a taste for sweets, carrot cake included. I find using canola oil to be better.
Usually it can be, yes.
a baking cake
yes
Flour, eggs, sugar, oil, flavoring, baking soda.
Applesauce or maynonnaise Applesauce or maynonnaise
If the recipe calls for oil, any vegetable oil is fine, though I don't recommend olive oil for that - it's distinctive flavor will come through in the cake. But corn oil, safflower oil, canola oil, etc, are all fine.
yes
I suppose it would depend on what you are baking. I have used coconut oil to replace butter in my gluten free pineapple up-side down cake. I have not yet tried to use it for other baking, but it works beautifully in the cake recipe.
Baking-wise it will work,but the butter may taste better.
No but you can make American muffens with vegetable oil
Yep, vegetable oil is always the best to use when you are baking.
You will need the basic cake ingredietns (butter, sugar, flour, baking powder / baking soda, eggs, milk, oil, ect.), along with some colored sprinkles or sugar candy pieces to add the rainbow effect. It should be a white cake mix, so that the colors show up the best, too.