"Draw" is not typically used in this way - a better replacement for "put" would be "bring," like this: "I am trying to bring smiles to the faces of my family and friends" "Draw" would really only be used here if you meant you were literally trying to draw (with a pen/face paint/etc) a smile on someone's face

Yes but put is more common

Yes the first sentence makes perfect sense. If you replace 'put' with 'draw' it sounds like you want to actually draw smiles on your family and friends faces...is this what you want to say ? :)