Translation:Your last name is Wang.
You could think of the radicals as letters in some cases, but the traditional characters tend to be more representative of the word/phrase the character represents. Look at the evolution of 车. The traditional character looks quite a bit like a cart, which was its original meaning, and now the simplified version cut a few strokes and it's a bit easier to write and remember.
Why would you not add 是 so it would be 你姓是王? And if it isn't grammatically correct, why is it not?
If 姓 is acting as a verb, then what you've written is "you are surnamed is Wang", rather than "you are surnamed Wang".
If 姓 is acting as a noun, then what you've written is "you surname is Wang", rather than "your surname is Wang".
As far as I know (and I may be wrong), it is possible to say 你的姓是王 (your surname is Wang) but it's not how it is normally phrased in Chinese, similar to how "you are surnamed Wang" is not how it is normally phrased in English.
It can be the usual way i.e. left to right and top to bottom; it can also be top to bottom and left to right (so vertical columns). I have never seen it right to left except in Chinese calligraphy, especially those displayed over shops e.g. restaurants containing sort of wishes for the store to do well e.g. 生意兴隆。