Final answer:
All three statements regarding the Speaker of Lok Sabha's powers, the automatic dissolution of the Lok Sabha, and the tenure of the Speaker post-dissolution are correct.
Step-by-step explanation:
The statements that are correct are 1, 2, and 3.
The role of the Speaker of Lok Sabha in India's parliamentary system is multi-faceted, including the authority to adjourn the House sine die or indefinitely. However, prorogation, which ends a session of the Parliament, is a power vested in the President of India. Following prorogation, only the President can summon the House of Parliament to meet again. This confirms the accuracy of statement 1.
Moreover, the Lok Sabha, the lower house of India's bicameral Parliament, is established for a term of five years. Unless dissolved sooner or the term is extended under special circumstances such as a national emergency, the Lok Sabha dissolves automatically at the end of its term, without any formal order from the President. This automatic dissolution by efflux of time substantiates statement 2.
After the dissolution of the Lok Sabha, the Speaker does not vacate their office and continues to hold position until just before the first meeting of the new House, ensuring continuity of authority and overseeing the transition. This continuation of the Speaker's role post-dissolution is in accordance with statement 3.