What is the story of the frog with Abraham (peace be upon him), what was the animal that threw wood onto Abraham's fire, and what is the ruling on killing them?

A hadith is narrated in Musannaf Abd al-Razzaq from Aisha (may Allah be pleased with her) that the Prophet (peace and blessings of Allah be upon him) said: "The frog used to extinguish the fire for Abraham, and the gecko used to blow on it. So he forbade killing this one and commanded killing that one."

The prohibition of killing frogs has been authentically reported in other hadiths, including what Imam Ahmad and others narrated from the Prophet (peace and blessings of Allah be upon him) that he "forbade killing ants, bees, frogs, shrikes, and hoopoes."

As for the gecko, it is authentically reported in Sahih Ibn Hibban that the gecko used to blow on the fire of Abraham, so the Messenger of Allah (peace and blessings of Allah be upon him) commanded its killing.