The correct option is 2 and 3 only.
Explanation
The Governor functions as the constitutional head of the state as well as an agent of the Centre. The provisions regarding the appointment, tenure, and powers of the Governor are enshrined in Articles 153 to 162 of the Constitution of India.
Statement 1 is Incorrect:
The Governor is not elected, either directly by the people or indirectly by the State Legislature. Under Article 155, the Governor is appointed by the President by warrant under his hand and seal. The Constituent Assembly opted for the appointment method to avoid potential conflicts between an elected Governor and an elected Chief Minister.
Statement 2 is Correct:
The Governor enjoys more discretionary powers than the President. While the President primarily exercises situational discretion (arising from political circumstances), the Governor exercises both constitutional discretion and situational discretion. Article 163 explicitly provides that the Governor acts on the aid and advice of the Council of Ministers, except in matters where the Constitution requires him to act in his discretion. There is no corresponding provision for the President in Article 74.
Statement 3 is Correct:
The Governor is appointed by the President, acting on the advice of the Union Council of Ministers (Central Government). While the Supreme Court has ruled that the office of the Governor is an independent constitutional office and not an employment under the Central Government, the method of appointment effectively makes the Governor a nominee of the Centre.
Key Takeaway:
The Constitution explicitly grants discretionary powers to the Governor (Article 163), a provision absent for the President. The Governor is an appointed authority, distinct from the elected nature of the President and the State Legislature.