"Why don't you smile her back" should be "Why don't you smile back at her"

1. He must be a bully when he was a kid. "He must of been a bully when he was a kid" 2. Her paintings are superb. (Correct) 3. She doesn't have a single clue on what he's saying. (Correct) 4. My heart shattered when I knew we were leaving the country immediately. (Correct, but no offten said that way. When I hear it I feel like the immediately should be placed somewhere else or shouldn't be there, but it is correct)

It seems right! I am a native english speaker but I have been studying French for 6 years. In english we would say new man, but in french it is kind of flip-floped they say the noun first before describing it. For example: In english "Pink skirt" and in French it would be "Jupe rose"

In general your letter is very well done! I am a native English speaker! Judging your letter it is pretty formal so I would make my speach in the letter a little more formal... If this letter isn't supposed to be formal then it is good as it is! At the beginning of the letter when you introduce yourself instead of writing "I'm John Baker from Vilnius Užupis Gimnasium in Lithuania" I would write "My name is John Baker and I am from Vilnius Užupis Gimnasium in Lithuania". You could put a "The" at the beginning of the "First Reason" sentence. If you begin your sentence with just a "First reason" it makes the sentence incomplete so the correct term would be "The first reason". In the same sentence you are speaking as if your are already part of the project (It seems that you are asking to join the project.) "The first reason why we would like to take part of your project is..." seems to be accurate. The first sentence in the secound paragraph the "We have" should be replaced with a "We've" would be more appropriate. In another sentence when you say "some kind of new adventure" it would be better to replace it with "some new kind of adventure." Later in your last paragraph in your question "How many students could arrive to your school? " It should be "How many students could I send to your school?" Or "How many of my student can visit your school?" The first one is more formal so I would chose that instead. Overall your english is excellent, there were no spelling errors, and there were no problems with any past tense or future tense. I wish you luck on your letter!!!