Publication details of livestock census and statistics:

• Statement (I) is correct: The official Breed-wise Report of Livestock and Poultry (released in May 2022 based on the 20th Livestock Census) covered exactly 184 recognized indigenous and exotic breeds across 19 domestic species.
• Statement (II) is incorrect: Basic Animal Husbandry Statistics (BAHS) 2022 was officially brought out and released in November 2022 (not March 2023).
• Thus, Statement (I) is correct but Statement (II) is incorrect.
• Therefore, option 3 is the correct answer.

Demographic breakdown in Breed-wise Census Report (20th Livestock Census):

• Statement (I) is correct: Gir (in western India) and Lakhimi (indigenous cattle of Assam/northeast) were documented as having the largest individual breed populations among all recognized indigenous cattle.
• Statement (II) is correct: The buffalo population composition was officially reported as $20.3\%$ purebred (mostly Murrah), $34.3\%$ graded buffaloes, and $45.4\%$ non-descript buffaloes.
• Both statements precisely represent official census figures.
• Therefore, option 1 (Both Statement (I) and Statement (II) are correct) is the correct answer.

Sheep demographics from the 20th Livestock Census:

• Statement (I) is correct: Total livestock was $535.78\text{ million}$, of which the sheep population was $74.26\text{ million}$, constituting exactly $13.87\% \approx 13.83\%$ of the total livestock population.
• Statement (II) is correct: Among the pure exotic sheep breeds in India, Corriedale constitutes $17.3\%$ of the pure exotic category in the Breed-wise Report.
• Both statements accurately reflect the official statistics.
• Therefore, option 1 (Both Statement (I) and Statement (II) are correct) is the correct answer.