The question focuses on the Home Rule Movement in India, which was a significant movement for self-governance led by prominent leaders like Lokmanya Tilak and Annie Besant. Let's evaluate each statement given in the question to identify which ones are correct:

1. Statement 1: "The advocates of the Home Rule Movement believed in constitutional methods and were opposed to violence and revolutionary action."
   • This statement is correct. The Home Rule Movement was largely based on constitutional means and aimed to secure self-government within the British Empire through non-violent methods. Both Tilak and Besant promoted legal and peaceful ways to achieve this goal, distancing themselves from militant and revolutionary actions.
2. Statement 2: "In 1917, the two Home Rule Leagues of Lokmanya Tilak and Annie Besant worked in cooperation with each other."
   • This statement is also correct. Although there were initially two separate leagues—Tilak's Home Rule League and Besant's All-India Home Rule League—they collaborated closely for the common cause of self-government. Their cooperation symbolized a unified effort towards the movement’s goals, gaining substantial public support during this period.
3. Statement 3: "Annie Besant confined her activities to the Bombay Presidency and the Central Provinces and the rest of India was left to Lokmanya Tilak."
   • This statement is incorrect. Annie Besant's efforts were pan-India, and her influence extended beyond the Bombay Presidency and the Central Provinces. She worked actively in various regions, and both leaders did not explicitly divide the country among themselves.

Based on the analysis above, statements 1 and 2 are correct, while statement 3 is incorrect. Therefore, the correct answer is:

1 and 2