The Use of the Present Perfect. What is natural?

There are quite a few factors at play here and it's not so easy to find the right way sometimes. The problem is even more difficult than native speakers may imagine as they don't understand how it feels when translating two different sentences, like "I have sent it to you" and "I sent it to you", you get the same sentence in Russian for each of the two.

As I understand it

1) I've sent you the letter. = 1) The letter is on its way to you now. 2) I am telling you a piece of news.

2) I sent you the letter. = 1) The letter might not be on its way to you any longer, because it happened when I was living in London and it was 2 years ago. 2) It may mean a specific situation in the past. "Do you remember us visiting the post office in the morning today. I want you to know that when we were there I sent you the letter.

There others nuances exist as well.

(My first language is the same as yours)

Have you meant? or Did you mean?

Definitely you would say “did you mean … ?” or “do you mean … ?”

I've sent you the letter and I sent you the letter.

They are the same. The former is British English, the latter is American English. In Britain I would say “I’ve sent you the letter” and in the Americas I would say “I sent you the letter,” but what I’m really saying is “I[’ve] sent you the letter.” The “have” has been contracted even further until it is non-existent. There are a lot of shorthands and omitted words in American English.

One thing to remember is that English has no rules, only conventions, and the conventions vary from place to place. From what I understand, it helps to laugh about it a little bit while you’re learning.