If I play a card and playing that card stuns me, do I get to use the effect of the card still?
The way I interpret the rules is that you can use the zap. Here's what the rulebook says when being stunned:


  Discard any or tokens you may have saved on your Player Board
  from other Heroes’ turns.If it is your turn, you may still play cards and take actions with whatever you have remaining after being Stunned.


The above was found on page 18 of the rulebook regarding Hero Health. View PDF Rulebook

I would assume that the stun happens immediately and you would follow the stunned steps from top to bottom.