I hope my questions make sense, but basically: why are there so many princes/princesses in Tolstoy, Dostoyevsky, etc etc. and are they what the English would consider to be princes/princesses? Or are these royal titles just the result of translation to English?
It is a translation of Кня́зь (Romanized "Kniaz"), a general noble title. Much like how individual noble leaders in the Holy Roman Empire or of various Italian city-states during the renaissance are called "Princes", it is a sort of generic noble title rather than indicating a connection to the immediate family of the Tsar. Think of it as Prince in the sense of Machiavelli's "The Prince", not England's "The Prince of Wales".
"Prince" as in "direct male descendant of the Tsar" would translate as царевич (Tsarevich), "son of the tsar" or Великий Князь (Velikii Kniaz), "grand prince" in later centuries. The full title for the Heir Apparent would be Наследник Цесаревич (Naslednik Tsesarevich)