How many Mythos cards can be played during the investigator's turns?
The keeper can play one Mythos card per investigator's turn.

From the rules, bottom of the first column on page 16:


  The keeper may play a 
  maximum of one Mythos card per investigator turn and may 
  never play Mythos cards during the keeper’s own turn.


Each player/investigator has his own turn each round.  Playing against 4 investigators, the keeper could play a maximum of 4 Mythos cards a round.

From pg. 6


  Playing the Game
  
  Mansions of Madness is played over a number of game rounds. 
  During each round, each player resolves his turn (starting with 
  the investigators)