Is it permissible for the husband to marry another woman before consummating the marriage with the first wife, and does she have the rights of a wife with whom the marriage has been consummated?

Allah Almighty permitted polygyny for men, with a maximum of four wives, provided there is justice between them. He did not stipulate justice in other matters. Therefore, it is permissible to marry a second, third, or fourth wife before or after consummating the marriage with the first, as long as the man is physically and financially capable of upholding justice. As for your sister's problem with her husband, the Sharia courts will handle the judgment.