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The correct answer is option 2: 1 and 3 only.
Statement 1 is correct. The Ain-i-Akbari is the third book in the Akbarnama, which was written during the reign of the Mughal Emperor, Akbar.
Statement 2 is incorrect. Abul Fazl was the author of all three books of the Akbarnama, not just the first two. Akbar himself did not contribute to the writing of the Akbarnama.
Statement 3 is correct. The Akbarnama and the Babarnama are indeed the best known accounts of illustrated Mughal official reports. These books provide valuable insights into the reigns of the Mughal emperors, Akbar and Babur respectively.
Therefore, the correct answer is option 2: 1 and 3 only.
Note: It is important to mention that the information provided in the given answer is generally correct, except for the mistake in statement 2.