Explanation:

• The President appoints the Prime Minister, but in practice, the appointee must command the confidence of the majority in the Lok Sabha: This is Correct Though the President has the formal power to appoint the Prime Minister, in a parliamentary democracy like India, the leader of the majority party or coalition in the Lok Sabha is typically appointed as the Prime Minister.

• The President can dismiss the Prime Minister at his discretion without any reasons:  This is InCorrect. The President cannot dismiss the Prime Minister arbitrarily. The Prime Minister holds office as long as they command the confidence of the Lok Sabha.

• All decisions taken by the Union Cabinet are communicated to the President through the Prime Minister:  This is Correct. This is in line with the principle that the Prime Minister is the primary channel of communication between the President and the Council of Ministers.

• While the President can send back a bill for reconsideration to the Parliament, if the bill is sent back to the President a second time, he is bound to give his assent: This is Correct. This provision ensures a balance of power, making sure that while the President can ask for reconsideration, the ultimate decision lies with the legislature if they pass the bill again.

Given the evaluation, options 1, 3, and 4 are correct.