No, your cake will have a strange texture. You can substitute mayo or apple sauce for oil though. You could even use butter, but I am guessing you are trying to get a low fat recipe?
a baking cake
yes
Flour, eggs, sugar, oil, flavoring, baking soda.
Applesauce or maynonnaise Applesauce or maynonnaise
It can be done, but it's not the best. The texture will be different, and the taste will be extremely off.
yes
There is moisture within the cake batter before it is baked. This comes mostly frow water, but also from milk and eggs and oil. As the cake is baked, the heat rises. This causes the moisture in the cake batter to turn to steam. Steam rises, but there is no place for it to escape. This causes the cake to puff up. Also, when the liquid cake batter cooks into a solid, it does naturally expand slightly, too.
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.
No.
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.
Baking-wise it will work,but the butter may taste better.
No but you can make American muffens with vegetable oil