English tends to live and die by word order. When that gets complicated things can get messy. The instinct in those cases should be to simplify.
In your case, neither of your examples works because of word-order issues.
Who's going to answer the Chemistry teacher's, who, I know, is beloved by Polinna, question?
Here you have separated the possessive noun ("teacher's") from its object by some intrusive yet unnecessary information. The ear wants you to make "question" follow on right after "Chemistry teacher's"—"Chemistry teacher's question"—and only then address the love interest of a student, if indeed that is who Polina is.
Who's going to answer the Chemistry teacher's question, who, I know, is beloved by Polinna?
Here you are asking us to believe that Polinna is somehow in love with the teacher's question; the word order has failed spectacularly.
Again, simplification is the key to fixing the situation. One example:
The Chemistry teacher, beloved by Polina, has a question. Who'll answer it?