I'm quite sure that I can say the Onegin's score meaning the opera “Eugene Onegin”. However, i feel that it's wrong to say the Onegin's leg meaning the character Eugene Onegin and that the correct form is Onegin's leg. Do I understand correct? Does that mean, that we can use the + possessive only if we talk about inanimated concepts?
A phrase like "The somebody's thing" means that the 'thing' is owned by "The Somebody". The definite article is applying to 'somebody' not 'thing'.
So you can say "The President's leg", meaning a leg of "the President" (it's implied that there is only one president by the context). "The Onegin's Score" would have to be the score of "The Onegin", and since "The Onegin" shouldn't be used, the rest shouldn't either. The same applies to legs.
Correct English would be "The score of Onegin". or "The leg of Onegin"; also "Onegin's score" or "Onegin's leg". You can also say "The Onegin score", which may sound the same, meaning the score called Onegin.
If the definite article is part of the title, then you can say it - for example: "The Flying Dutchman's score".