Hi Dna,

Your notebook is totally understandable however I would like to make some comments. I wont correct all of it because I believe someone else, (preferably a native) will do it. If no one will, then I will try to write more in detail.

I know that in Russian, the order of the words doesnt matter much but in English, I recommend that you use SUBJECT+VERB+OBJECT. You can put time information to the beginning or to the end
...my cousin lived in such an apartment for more than 30 years
For more than 30 years, my cousin lived there

And dont forget am/is/are :)

She was lucky, in that sense, that their apartment consists of only 3 rooms, two of which are belonged to her family and the third one was belonged to her neighbour.

Living with such a person, in principle, is very difficult, and that is even more difficult when that person is not your relative, but only a neighbor.
Few years ago, when this old woman became too old, her relatives took her to their home to take care of her. And her granddaughter ...

Communal Apartment

Do you know what a communal apartment is? Communal apartments became a symbol of the Soviet era in Russia. Communal apartment is kind of a mini-dormitory. Many families would settle in one apartment of 3-8 rooms. Or- There would be 3-8 rooms in one apartment, and each room would have a separate family. The toilet, bathroom and kitchen would be shared. Such apartments in the past belonged to the rich people. (Really?!)
My cousin lived in a communal apartment for more than 30 years. She was lucky, in that sence, because their apartment consistsed of only 3 rooms, two of which belonged to her family and the third belonged to her neighbour(American English spells nieghbor). And, according to the classics of the genre (not a common saying), a neighbor, of course, was a harmful old woman. (And, of course, the nieghbor was a harmful old woman. Better.) To live with such a person, in principle, is very difficult, and it is even more difficult when a the person is not your relative, but only a neighbor.
A few years ago, when this old woman became too old, relatives took her to their home to care for her. Her granddaugher moved into her old room with her husband, two kids and a cat. The new neighbours were polite and quiet as much as they could be.

But a Anyways, to live together in such a big community is really difficult. You have to share a bathroom and a kitchen when everyone is going to work or school in the mornings, and then when everyone returns home, cooks and relaxes in the evenings. (This is a long sentence and can be written many different ways.)

In general, I've spent a lot of time and effort to pass the “seven circles of hell” of the Russian bureaucratic system. My cousin's family got a small amount of compensation money from the state, added their own money and had agreed with the neighbors to buy their room. Now the whole apartment belongs to my cousin and her family. She gave birth to a child and makes repairs in the “new” apartment. And I’m very happy for her.

You wrote this very well, and if you had read this aloud anyone who speaks English would understand what you were saying.