"Do native speakers use the word 'shall' in English?"
There's this funny myth being taught in Russian-speaking countries (Not sure how I should word that - I do realise you're from Ukraine) that native English speakers don't use "shall" anymore. Well, what an absolute lie that is. :)
We certainly do use shall in offers or suggestions such as "Shall I phone you this evening?" This is elementary level. Using another modal changes the meaning.
As well as the daily use of shall, once you get past intermediate level, you will definitely discover the other functions of shall, and realise that you do need to know how this modal works.
My advice? Get a better textbook... written by a native English teacher. There are too many other fictions and fake rules being taught to students, and this "obsolete shall" is just the tip of the iceberg.