What is the authenticity of this supplication: "O Allah, I entrust to Your keeping my religion, my trust, my offspring, my wife, my heart, the hearts of my offspring, the heart of my wife, their hearing, their sight, their private parts, and all their limbs"?

There is no harm for a Muslim in supplicating with "I entrust you to Allah, with whom deposits are not lost," because there is evidence for this in the Prophetic Sunnah. It has been authentically reported that the Prophet, peace and blessings be upon him, said: "Indeed, when Allah is entrusted with something, He preserves it." The Prophet, peace and blessings be upon him, used to bid farewell to a traveler by saying: "I entrust to Allah your religion, your trust, and the last of your deeds."