When does the next player start their action?
#1 is correct. 

I can't find a place in the rulebook that makes this super-obvious/clear, but there's this:


  The game then proceeds clockwise,
  from player to player. Once all players have had a turn, a new
  turn begins. 


And:


  During his turn, each player must now either:
  
  
  Expand the reach of his race through new conquests
  OR
  Put his race In Decline to select a new one.
  


Basically, all the parts of a turn that are described are what one player does on his own turn, before the game moves to the next player's turn.