"The library is very quiet."
No. Sono is never 'the' duolingo has many huge problems with their new Japanese program and this is one of them. Japanese does not have articles EXCEPT demonstrative articles. If you don't remember highschool English...those are 'this that and that over there'. Sono means 'that' and that is ALL that it means when it isn't in an expression.
No, その means the in the sense that it is referring to something talked about in the previous conversation. It doesn't need to be physically near to the speaker and the listener.
If you go through the English course for Japanese speakers every "the" translates to その so it does not relate to the beta.