1. What happens is a gate appears on a space a player is currently on?
The rules are unclear on this, so by inference I play that the investigator is immediately sucked to the Other World, just as if they uncover a gate as a location encounter.
2. What does happen when you don't have any money and lose your Strength/Sanity points in Arkham?
The location descriptions for the Hospital and Sanitarium cover this -- you pay what you can, get one turn's stay, and then must leave.
3. How do flying monsters move if: a.) there are no players in Arkham?
Unclear. I play that they just sit there.
b.) All players/the nearest player is inside a building?
Also unclear. I play that they lurk outside the door, but you could easily argue from the Flyer glossary entry that they don't move unless there's someone on the sidewalk.