I'm not trying to confuse things here but I'm not 100% sure you can use Elemental Fist or Stunning fist as part of Flurry of blows. Each of those implies that you make an unarmed strike. An unarmed strike is a standard action. FoB is a full round action. I understand that FoB is a number of unarmed strikes and would rule that it could apply to any of the attacks in a FoB situation. But I would also rule that monks with both elemental fist and stunning fist could use one or the other in a FoB.
There is way to use both in a round but not really with FoB. A Monk of the 4 Winds that has both could use his or her Slow Time ability to get 3 extra actions, and he could then make 1 elemental fist attack and 1 stunning fist attack.
As for the elemental fist trasfering to weapons, even monk weapons would be why? Base damage for weapons does not go up like a monks unarmed combat damage does.