Fried carrot cake, white and black version

That's right. The black version has sweet black sauce added. Yes it's called fried carrot cake in Sg 😄

I'm not sure what your are referring to. For this kind of carrot cake, few will make it at home.

That would be 萝卜糕, as they use white radish(大根), which is known as 白萝卜 in mandarin