Please read my comments below the letter:

Letter for my English-speaking friend (for the exam)

Dear Mary,
Thank you for your letter. I hope everything's okay. Let me answer your questions.
I was upset to hear that your quarreled with your friend. Unfortunately, even real friends quarrel. But that is not a reason to break up. Recently, I quarreled with my friend too. She didn't help me with my project, althrough although she promised to help. I was so upset, but then she came to me and ask me to forgive her. I was amazed. Usually I'm the first one to apologize when we quarrel. I always say that we all face problems in daily life, and it's important to deal with them quickly and effectively. Friendship is one of the main things in our life. That's why we should be together and support each other.
And I want to ask you some questions. How many close friends do you have? Where does your aunt live? Are you going to your aunt with family or friends? Write me again.
Best wishes,
Ann

This is very well written. It sounds authentic. If I was your professor, I would give you a good grade :-)

I have some suggestions for you. I have unlined some words in your letter, see the following for my comments:

Real friends - try "good friends" or "best friends"

But - The "but" that is underlined, this word is a conjunctive or a connector. Sentences should not start with a conjunctive. So you could say, "Unfortunately, even good friends quarrel but that is not a reason to break up." Or you can say, "Unfortunately, even good friends quarrel. This is not a good reason to break up." The way you have written this thought is okay. The examples I have given you are better grammar.

Be together - try "stay together"

Are you going to your aunt with family or friends? - What are you trying to ask? As the reader, I do not know what Mary said about her aunt. You can clarify by saying "Are you going to your aunt's house with your family or friends?" or "Are you going to visit your aunt with your family or friends?"

GOOD LUCK!