Is the Hadith that states: "Whoever kills a gecko with the first strike will receive one hundred good deeds" authentic, and if it is, why was the gecko specifically mentioned?

The hadith you inquired about is authentic, and its wording is: "Whoever kills a gecko with the first strike, a hundred good deeds will be recorded for him; with the second, less than that; and with the third, even less than that." It is narrated by Al-Tirmidhi, Ibn Majah, Ahmad, and others. The reason for the increased reward for killing it with the first strike is to encourage prompt action in killing it, so it does not escape. The reason for specifically mentioning the gecko is that it is one of the fuwaysiqat (small evildoers) that cause harm. The Prophet, peace and blessings be upon him, commanded their killing both inside and outside the sacred precincts (Haram), and he called it a fuwaysiq. The Prophet, peace and blessings be upon him, mentioned five fawasiq (evildoers) that may be killed both inside and outside the sacred precincts: the mouse, the scorpion, the crow, the kite, and the rabid dog.