Is it allowed for the next player to take the next tile while the current player hasn't finished his move yet?
Tom Sloper, an apparently very experienced American player, writes about etiquette. It turns out that "One person's bad etiquette is another person's official rule" - so you have to adapt. There is a huge difference between tournament play and casual games with friends. Also, you need to consider house rules. In the end, there is a definite answer to your question in the rule book, but its usage in practice depends on the circumstances.


  DISCARDING AHEAD -- When I play with my Japanese friends, they yell at
  me if I claim a discard and then make the exposure before discarding
  (nobody can play until I discard, and they want to PLAY!). But when I
  play with my American friends, they call me dead if I discard before
  making the exposure. One person's bad etiquette is another person's
  official rule! Why it's a bad habit -- Do it with players who strictly
  follow the rules (like in a tournament) and you'll be called dead.
  
  PICKING AHEAD -- Many American players think "picking ahead" (AKA
  "playing with a future") is the correct way to play. But when these
  players come to a tournament... watch out! Old habits die hard. And
  picking ahead is the surest way I know to get called dead in a
  tournament. In case anybody's been living in a cave, picking ahead is
  a no-no. That other mah-jongg association (I'm not naming names, but
  their initials are NMJL) HATES picking ahead! Read their rulebook and
  yearly bulletin sometime if you don't believe me. Why it's a bad habit
  -- (1) It causes too many conflicts when someone wants to claim a discard. You might think that it makes the game go faster, but it
  really doesn't. (2) Do it in a tournament and you'll be called dead.


He adds a few more points to consider (including "avoid making everyone wait"), it's a very interesting read.