Is it permissible to dedicate supplications (Du'a) and ذکر (Dhikr) exclusively to oneself and one's family, such as saying: "Our Lord, grant me, my children, my siblings, and my husband good in this world and good in the Hereafter, and protect us from the torment of the Fire" and "We, my parents, my siblings, and my husband, are pleased with Allah as Lord, with Islam as a religion, and with Muhammad as a Prophet, Messenger, and Intercessor"?

The best approach for a Muslim when supplicating is to pray for their relatives and all Muslims. You have done well by including your relatives in your first supplication. As for other dhikr such as "I am pleased with Allah as my Lord" and "O Allah, I bear witness to You," these are prescribed dhikr in the first-person singular only. It is not permissible to include others in them, as they are declarations about matters that only the speaker is privy to.