Is what the girl's family did permissible by demanding a divorce due to inequality of lineage, and does divorce by coercion take effect? And is it possible for the husband to take back his wife without her family's knowledge after this divorce which occurred before consummation?

The girl's family is mistaken in asking for her divorce from you and in pressuring her father. This is an act of corruption that the Prophet, peace and blessings be upon him, forbade when he said: "He is not of us who incites a woman against her husband." Your divorce is valid unless there was lawful coercion. If the divorce occurred before consummation but after valid seclusion, you may take her back as long as she is within her waiting period ('iddah). It is best to refer the matter to the Sharia court. Piety and good character are the considerations for compatibility.