Does John Pain get cards drawn at the beginning of a player's turn or from Stage Coach and Wells Fargo?
No, in this case draw! refers to flipping a card and checking the suit revealed for something to fire like barrel or dynamite. Players drawing cards for their turn, stagecoach/wells fargo/dodge/bible or any other ability that adds a card to hand does not count.

Here is a guide that gives some more information on the draw! ability.

http://bangcardgame.blogspot.com/2011/09/character-guide-john-pain.html