I'll try to answer what I can.
1 - I think that the Government initially rejected Olav because they thought his ideas were crazy, so he worked alone on his research and later Nelson found out his discoveries were actually true.
3 - One of Olav's recordings that Nelson uncovered had Olav summoning the hidden people with a song.
6- I think the Moon.
7 - Davner was locked in the factory because the Brotherhood claimed the Hidden People whispered to him, and they wanted to keep him contained without him escaping so they could use him for their own purposes.
9 - Mrs Garett was trying to help protect Nelson because The Brotherhood and Sherif Bahg were angry at Nelson for interfering with their plans to keep Davner locked in the factory, and if they knew Nelson was at the Inn, they likely would have harassed him somehow.
10 - Yes
12 - I think the Hidden People told Nelson to go to the Lunar Ray, and the scene in Space was Nelson hallucinating while he was at the Lunar Ray. I don't think Nelson intentionally wanted the Hidden People to speak to him, but I could be wrong.
13 - I think the Government wanted to keep the failed flight of the Hermes II and the Lunar Ray a secret, so they tried to convince the world that Davner was dead when he was actually involved with the conspiracy, but again I could be wrong. To answer your question, after the mission.
14 - It was both. It was the overall conspiracy and the clues coming together that made the government come.
15 - The Hidden People attracted Alfred.
16 - Yes, the Astronauts are bad, but only because of the Lunar Ray.