Corrections in red.  Stylistics in blue.

Could you check this?

1. You didn't forget to send the letter which I gave you? Having put a hand in his pocket, he found there the letter which he had forgotten to send. As soon as I read the first lines, I remembered that I had already read these words somewhere. I absolutely forgot that someone had borrowed someone the dictionary, and I looked for it on the shelves for a long time. 2 . The case was serious, and the doctor suggested a consultation be arranged. Seeing our difficulties, he offered to help us out. 3 . Whether it kept the promise to be engaged in these business? [2 ?? ] He promises to soon become one of the best chess players in our club. 4 . In chapter 2 the author continues to describe the economic situation of the country. In chapter 2, after a short analysis of the national economy, the author describes a modern political situation in it.  5 . Its art couldn't but cause admiration. Whoever saw his pictures, couldn't but admire its art. [3]

[1]  When translating the Russian word "опустить" in the context of a letter, it's better to simply say "send".  Exception:  When the sentence contains the word "mailbox".  Then could say, "Drop the letter in the mailbox." [Опустите письмо в ящик.]

[2]  Sorry, I couldn't understand the sentence "Whether it kept the promise......"  

[3]  The two sentences numbered "5" could possibly be correct in certain contexts which I can imagine.  But they are not entirely clear without a context.