Is the widespread saying: "Teach your young children Surat Al-Fatihah, for they will recite it throughout their lives, and Insha'Allah you will be rewarded for it," correct, and will the one who teaches a child receive a reward for their recitation of Al-Fatihah throughout their lives?

Yes, the statement that a Qur'an teacher receives the reward of everyone who recites the Qur'an because of him is correct. This is indicated by the hadith (definition): "The one who points to good is like the one who does it," and "Whoever calls to guidance will have a reward similar to the rewards of those who follow him." The teacher of the Qur'an to children receives a reward similar to theirs for the duration of their recitation, and a reward similar to the recitation of those who learned from whom they taught. For reciting a single letter of the Qur'an, there are ten good deeds.