There are several different types of academic writing OR Several different types of academic writing are existed (which of thisese two sentences would be more appropriate for academic style?).

Either is fine, but I prefer the first.

I think that I a'm using referencing writing right now, because I'm doing it myselfout of my own will and not for ato takinge any assessment, but. It's only for remember someing pieces of information and developing my writing skills.

I'm actually not sure what "referencing writing" is and I couldn't find anything on it. You may want to research this a little (Russian to English Wikipedia) to see if you're using the right word.

Alt: I think that I'm using referencing writing right now, because I'm doing it out of my own will to remember pieces of information and develop my writing skills, not to take an assessment.

From this, I realize that all my texts here are just a reflection of what I ha've read about academic writing without using academic language.

Sometimes my teacher asks me about writing answers for some short questions, or criticizing an article concisely and briefly.

Since you have "sometimes" I think saying "some" sounds redundant.

Also, did you mean "to" instead of "about"? This changes the meaning of the sentence. If she's asking you to do those things, that means she's requesting that you do them. If she's asking you about them, that means you've already done them and she's asking you for clarification.

I guess that both those tasks are very useful for checking my knowledge about a particular concept.

You could also use "pretty" instead of "very" here. Since you say, "I guess" it sounds like you're unsure and "pretty" is less strong than "very" is.

On other days my teacher gives me tasks to prepare well-structured essays based on references and quotations, which I write oin my native language successfully.

The first type is an argumentative essay which persuades the audience that particular point of view is correct.

A good argumentative essay demonstrates understanding of the topic, where the author uses a wide range of sources, critically analysizes them, and provides clear arguments.

alt: "...where the author uses a wide range of sources, critically analyzes them, and derives clear arguments from them."

This would indicate that all the actions are linked to the sources.

A descriptive essay is another type of essay, which demonstrates a strong understanding of subject.

I think adding "of essay" adds a little bit of clarity, but either is fine.

An evaluative essay can be very challenging and students should to be preparinge to compare and contrast all sides of a topic.

ItThis means that author should analysesze all aspects of the essay and decides which he or she agrees with.

analyze = US; analyse = UK

Alt without "should": This means that the author analyzes all aspects of the essay and decides which he or she agrees with.

I hope one day I can do ithat in English successfully.