Is it permissible for male and female students to communicate via messages to solve assignments or to discuss urgent study matters, while a third party is privy to the conversation without the knowledge of the other party who might be bothered by that? And is the approval of the girl's family for the communication sufficient, without them being informed of the content of the messages?

It is not permissible to speak with a non-mahram woman on the phone except for a legitimate, recognized need, and on condition that she does not speak in a soft, alluring voice. When speech is permissible, there is no need to let others hear the call. If there is no legitimate need, he should not speak to her, nor should she speak to him, due to the presumption of fitna (temptation/strife). The need of female students for an explanation of lessons is not considered a need that necessitates conversation, as they can ask the subject professor.