What is the ruling on conversations between an engaged woman and her fiancé with the knowledge of her family, and what is the ruling on her insistence on marriage despite the refusal of her younger brother who lives far away?

The suitor is a stranger to his fiancée before the marriage contract, and it is not permissible to speak to him except out of necessity. If the suitor is suitable, your brother has no right to prevent him; if he does so, he is an oppressor (عاضل), and guardianship transfers to the more distant guardian or the ruler. It is advised to reach an understanding with your brother and convince him. If that does not work, there is no harm in marrying under the guardianship of another guardian or raising the matter to the Sharia (definition) judge.