1) This carrot cake recipe can be made without egg. For the eggless version, use little extra yogurt or applesauce (1/2 cup instead of 1/3 cup).
2) If you wish to use only brown sugar you can skip white sugar. But make sure the sugar granules are fine enough to blend well.
3) The use of nuts is completely optional. Nuts give a delicious crunchiness to this carrot cake.