How can I make a carrot cake with only a few of the required ingredients?

I want to make a carrot cake today, but I cba to go to the shop to get some of the ingredients that the recipes I seem to read require. I was wondering if someone knew a recipe for carrot cake (and icing) that only needed the ingredients I do have. Also there are some other questions I want to ask at the end.

Ingredients:
– white self raising flour
– eggs
– cinnamon
– carrots
– baking powder
– butter
– brown sugar
– vegetable oil
– vanilla extract
– castor sugar
– icing sugar
– salt

1. Why do some recipes include both bicarb of soda and baking powder. What would happen if you didn’t include the bicarb to the cake?
2. What does the wholemeal flour do to the cake that normal white flour doesn’t?

I was originally going to combine a few recipes together but I assumed that’ll end up in disaster as I am not that great with baking. Also if any integral ingredients are missing that detracts from the carrot cake experience let me know as I will just scrap this whole idea altogether. No-one wants to eat a sub-par carrot cake.

Thanks for the help.

P.S. I would prefer measurements in grams/ounces/tablespoons but not in terms of cups. Thanks.