Is it permissible for a young man and woman to chat via Facebook during the pre-marital period and the family's engagement, with a guardian (mahram) present sometimes, and what are the regulations for such a conversation?

It is not permissible for young men and women who are not mahram to get to know each other or communicate, whether online or otherwise, even under the pretext of marriage. This is because it is a gateway to evil and corruption that leads to dire consequences. Jurists have explicitly prohibited speaking with a non-mahram woman without necessity. Correspondence is no less serious than verbal communication. You must cease communicating with the girl via Facebook or any other means, whether her family knows or not. If you desire to marry her, then proceed with the marriage. Until you عقد عليها the legitimate marriage contract, be patient and adhere to the limits set by Allah. If there is a necessity for a call or correspondence after the engagement, let it be only to the extent of that necessity.