Is this method correct for seeking protection: gathering the palms, reciting the Mu'awwidhatayn (Chapters of Refuge), blowing into the hands, and then wiping with them over the children, the husband, and oneself? And does protection occur through this, or is it necessary to recite and wipe over each individual separately?

Protecting family and children can be done for them collectively. The Prophet, peace and blessings be upon him, used to seek refuge for Al-Hasan and Al-Husayn, saying: "I seek refuge for both of you with the perfect words of Allah from every devil and every harmfulthing, and from every evil eye." And when any of his family members fell ill, the Prophet, peace and blessings be upon him, would blow upon them with the Mu'awwidhat (Surahs Al-Falaq and An-Nas) or wipe them with his right hand, saying: "Remove the suffering, O Lord of mankind." It is recommended to blow during ruqyah (exorcism/spiritual healing).