What is the ruling on slaughtering puppies found by the roadside to prevent their death from hunger, thirst, or torture, given the inability to raise them?

Your friend must seek Allah's forgiveness and repent to Him for killing the small dogs without cause. He could have shooed them away if their barking bothered him, or left them to find food. Killing an animal without cause is forbidden in Islamic law, as evidenced by the Prophet's (peace be upon him) saying: "Whoever kills a sparrow or anything larger than it without its right, Allah will question him about it on the Day of Resurrection." Therefore, killing non-vicious dogs or harmless animals without cause is impermissible. There is no specific expiation for your friend other than repentance and seeking forgiveness.