Of your sentences only #1 is correct, however these sentences are correct: I will write an email to you. (You got your word order confused) I hope you write me soon. I hope you write to me soon.

You need to use "I'll write you an email." In other comments, I think there's been a misunderstanding between what is correct and what is understandable. This is how it works: "Write" can take two objects. A direct object (an email) and an indirect object (you). If we choose the order [verb] [direct object] [indirect object], then we use "to" or "for" before the indirect object. "I'll write an email to you." If we choose the order [verb] [indirect object] [direct object], which is also found in other European languages, then we don't need "to" or "for". "I'll write you an email." This does not work for all verbs, remember. If the verb can naturally take an indirect object, you can put it between the verb and the direct object, without any preposition.

the most natural way to say this is "I'll email you". Or, "I'll send you an email". I don't think a native speaker would ever say "I'll write you an email". We would say "I'll write you a letter", but sadly, hardly anyone writes letters anymore.

"write you an email" - good "write an email to you" - good but "write to you an email" - not wrong, but unnatural and unusual.

Both grammaticaly good, but ill write u an email is more natural