When we was asked to write or tell our full name, should we write/tell surname and name only or surname, name and patronymic name?
And also in what order we should tell them?
Well, we don't use patronymics in English names. However, some of us have a middle name, and we would understand your patronymic name as a kind of "middle name".
To give our full name, we would use first name / middle name / surname, in that order. "My name is John Winston Smith", for example. Saying "John Smith" is also OK, as we'd only include our middle name in formal situations, or for administration purposes. We would not normally say "Smith, John (Winston)" unless we were registering for something.
Tchaikovsky would give his full name as, "Pyotr Ilyich Tchaikovsky".