Today

Today was a really long and busy day.
Early in the morning I was woken up by a barking dog and I could not go to bed anymore. I then decided to have a breakfast but I had find out discovered that I didn't have any milk, so I had to run to the 24 hour shop. After I had my breakfast I walked my dog to the nearby park. I then took a shower and I went to work got out to go to my job.
The train was late by 10 minutes so I started to read a the newspaper. When I finally got at to my office I had a meeting with my manager and an outside person (an outside person sounds wrong - perhaps you can say: client, patron, customer etc). After that we had a lunch together at the a nearby restaurant. I had a greek salad, my manger got macaroni cheese and the other person got a pizza.
In the afternoon I had to work extra hard because my colleague was not there (away) and we had many things that had to be done.
When I left the office I was exhausted and I fell asleep on the train. When I suddenly woke up the train was just at my station.

That's pretty good writing. Just a few small spelling errors, and the main points are:

1) Revise some prepositions - "I get to my office" but "I arrive at my office".

2) In English we simply have breakfast or have lunch or have dinner, we don't put a/the before it. Good rule to remember. But when we are ordering or describing the actual food, you generally need a/the, which you did. I got a pizza, she'll have the pasta.

A man who lost himself

A man woke up in the middle of the night and sat on the bed. He looked out the window and saw the clouds drifting across the sky. It was a foggy November. It seemed that all his life was not so long as this autumn (sounds awkward - try rephrasing this). These timeless gray days were not going to end finish. The dim light of the moon illuminated the room.

Last days Lately he couldn’t sleep peacefully. During the day, this feeling was not so strong, but when the night came, he felt that some strange thoughts creep into his mind. He understood that if he didn’t find out what’s was going on, something terrible would happen.

Despite the cold in the flat it was hard to breathe. He laid down on the bed but he couldn’t fall asleep. It was impossible to sleep and he decided to go out and strolled around the city for an hour. He did so at times. The street was silent. He walked down the street. At the end of the street, he saw a light from some windows. It was a musical school, and at this moment he realized that he was late for a lesson. He sprinted to avoid the anger of the strict music teacher of music. He breathed frosty air and ran as fast as he could.

The music school was light and warm. The sounds of eternal timeless classical music came from all sides and merged into a symphony single cacophony. Then he saw the strict teacher, who came to him and said: “Get a move on! The time has come, show all of them what you can do!”. «Sorry, but I don’t remember how to play…” – the man said quietly. The man briefly looked up at the teacher but there was no one around, it was dark.. «Where are you?» - cried a the man in the darkness.

«Enough, calm down, son. It is too late. Mom must be worried. We’d better get moving». The man turned and saw his father, who was young and strong, his eyes sparkled and his mouth steamed. There was snow around the street. Snow gleamed under the light of the street lamp. «Sit on the sledge, son, I’ll take you home» - said his father with a kind smile.

The man was sitting on the sledge and was on the verge of falling asleep. “Do you remember how you argued with your mother today? You are wrong, your mother loves you.” – said the father softly. He said something else, joking and laughing. The man sitting on the sledge felt better than ever before. He almost didn’t listen to what the his father was telling him and just at times occasionally answered with a smile: «Yes, Dad… of course dad..»

I went to (the/my/our) new apartment.

I went to the/my/our new apartment, which was built this month.
It's equipped with an air conditioner, 2 stoves and a monitor intercom. 
I'm in charge of this apartment so I need to get tenants untill next month.

Good. Remember to use articles. You can't simply say: I went to new apartment, or It has air conditioner.

In English, the sentence you wrote means option 2. Ie. I can't tell her, because she is my sister. (meaning, if she were someone else, maybe I could tell him/her). If you want your sentence to mean option 1, you have to say: I can't tell her just because she is my sister. You need 'just' there. In this sentence, it means "Just because she is my sister, doesn't mean I have to tell her anything'.

A letter in reply

Dear Red,
I'm sorry to know hear that you are having such a bad time at the moment.
In your letter you said that you were so easy to lose your temper easily and that has made your normal daily life difficult. In fact , many people always had experience growing problems when they were are young. It's a normal phenomenon which that is not so hard to overcome, so don't lose faith be afraid of it.
Here are some suggestions on how to get over overcome it. Frist First, try to calm yourself down when you're going about to lose your temper. Second, if you have lost your temper to at someone, apologize to him or her, and afterwards then ,ask yourself the reason why, because it will clear your mind when you do that.
You are welcome to write to me when you have any trouble next time.
Yours,
Jamie
(这篇小作文是以编辑的身份写给读者的回信，帮我改一下吧，十分感谢！)