Are there any prophetic hadiths regarding raising children, and is the hadith: "Indulge them for seven, discipline them for seven, and teach them for seven" authentic?

The Prophetic Sunnah is rich with hadiths that encourage raising and educating children, such as: "Instruct your children to pray when they are seven years old, and beat them for it when they are ten years old, and separate them in their beds," and his saying to the one he taught table manners: "Say Bismillah (in the Name of Allah), eat with your right hand, and eat from what is in front of you," and his saying to another: "O young boy, I shall teach you some words: Be mindful of Allah, and Allah will protect you." As for the saying: "Play with your child for seven years, discipline him for seven years, and befriend him for seven years, then let loose his rope (i.e., give him freedom)," it is among the wisdoms narrated from the predecessors, and some have attributed it to Umar ibn al-Khattab, may Allah be pleased with him, but the authenticity of its attribution to him is questionable.