How can one deal with a stubborn, jealous sister who seeks attention, and whom the sister asking the question suspects of talking to a young man, while the denying sister rejects this and claims her family hates her?

As long as you are not certain about your sister's ill intent, it is not permissible for you to harbor ill thoughts about her. Rather, you should advise her generally by explaining the prohibition of a woman establishing a relationship with a non-mahram man, and the danger and consequences of such relationships. You should also try to connect her with righteous company that will guide her to the straight path, all with wisdom and gentleness, and be a good role model for her.