The out is not necessary, but the way I see it is that sometimes we put 'out' to emphasize the fact that the thing you are writing is being put on the page, as if it is spread out on the page for the eye to see. I could use it for example if someone tells me their name and I don't fully grasp how to spell it, I may say 'write it out for me, please'. I would ask that to be able to see it in writing. We also say 'I'll write it down' to emphasize that it is put down onto the page where you can find it later. This is used for things we don't want to forget, or that we want to find later. For example if I make an appointment with someone, and I know she is likely to forget, I may say to her 'How about you write it down on your calendar, so you won't forget"

I am writing it out. I am writing it Out is not necessary. They mean the same. Hamlet is the Kings nephew is a declarative sentence. Hamlet is the nephew to the king is passive. Both are correct , but usually we prefer declarative sentences as the make the point clearer in a more assertive manner.

'it out' is hard to explain. you probably understand the 'i'm writing it' part, but the 'out' just means you're going through the process of writing it. either of the 2 sentences make sense, but the first one is more natural and clear 'hamlet is the king's nephew'