My name is Lady Nora, people recognize me for helping people in need every time they're in need.

Sounds more natural, in my opinion. Your sentence was fine though :)

My super power is that I can fly, I have super speed, and I also have friends who are always there for me.

My friends are also superheroes; they a're called Luna, Alexsandar, Alex, Leon.

Unimportant note: "they're called" in the context of people is common in Britain, but in other places like the US, I usually hear "their names are ..." or "they're named ..." more often.

They are happy to help me, even if I likoften have to save the world from evil.

I wasn't sure what you meant. Did you mean something like "Although saving the world from evil so often is hard, my friends still help me"?

My biggest enemy is Mister Moon, hiswhose mission is to destroy the country and make himself king.

"his" was fine, but your placement sounded a bit unnatural.

But he doesn'tnever succeed at all, hs. He wears a black suit and a black cloak, which is how we recognize him.

WhenAs soon as the country is in danger, I come right away, I come because I know that (in) thisat way, I am helping myself and all the people who need help.

1. For the start, you could also write "Whenever the country is in danger, i come right away ..."

2. The "in" from "in that way" is optional, depending on the dialect some prefer to use it or drop it.

Of course, whenever Mister Moon comes to realize his plans again.,

"whenever" is used when talking about something which happens regularly, but which always has the same result, like "Whenever I drink coffee, I become alert."

T, together with my friends we triedy to stop him and if. Since he lives on the moon, then he's gotten the name Mister moon.

1. (comma added by me because this should continue from the last sentence.)

2. Your sentence is mostly fine, but saying "he's been named Mister Moon / he's been given the name Mister Moon." might sound a bit more natural.

My super -power is that I can fly, I have super speed, I alsoand I have friends who are always there for me.

My friends are also superheroes, they are called Luna, Alexsandar, Alex, and Leon.

They are happy to help me even if I like to save the world from evil.

even if sounds strange here, I'm not sure what you meaning is.

My biggest enemy is Mister Moon, his mission is to destroy the country and make himself king.

But he doesn't succeed at all, h. He wears a black suit and a black cloak, which is how we recognize him.

When the country is in danger right away, I come, I come immediately because I know that in this way, I am helping myself and all the people who need help.

Of course when, Mister Moon coames back to realize his plan again.

TogetherBut, with my friends, we tried to stop him and if. Since he lives on the moon, then he got the name Mister moon.