What is the proof that all Muslim children are in Paradise, taking into account the Prophet's (peace and blessings be upon him) hadith about the deceased child of the Ansar, and the story of the boy killed by Al-Khidr in Surat Al-Kahf?

Muslim children are, by consensus, in Paradise. More than one scholar has related this consensus, because they are not held accountable. However, this does not mean bearing witness that a specific child is among the people of Paradise, for the Prophet, peace and blessings be upon him, forbade Aisha, may Allah be pleased with her, from doing so, as one does not bear witness for a specific person, whether for Paradise or Hell.

As for the youth killed by Al-Khidr, there is a difference of opinion regarding him: was he an adult or not? If it is said that he was an adult, then there is no difficulty, and Al-Khidr, peace be upon him, would have known of his disbelief. If it is said that he was not an adult, then the disbelief of a discerning child is valid according to many scholars, but he is not killed in our Sharia until he reaches puberty. The term "ghulam" (youth) can be applied to an adult, and His saying, exalted be He, "without [taking] a life," is more befitting for an adult than for a child. It is possible that killing a disbelieving youth before puberty was permissible in that law, or because he was a disbelieving adult, or to repel his harm from his Muslim parents.

In summary, the evidence indicates that the children of believers are in Paradise, and there is a consensus on this. As for the children of polytheists, there is a difference of opinion regarding them, and the preponderant view is that they will be tested.