How are the stats recorded? Is he the quarterback?
im not an expert, but im not sure the punter throws the pass. i always thought it was given to the quarterback. again, i dont know what im talking about.
You are thinking of a place kicker, not the punter.
No. He is a punter who throws a pass. Stats are recorded as normal, I suppose. It would just be an aberration in the stats, just like a lineman scoring on a fumble.
What if the team re-positions and the punter comes under center or moves in the shotgun?
Then he is the quarterback for that play.
Tom Tupa has enjoyed a long professional career as a punter/quarterback.
From a statistical point of view, it makes zero difference. If a player throws a pass, he throws a pass–period. There is no distinction between “passes thrown by quarterbacks” and “passes thrown by other players”. Running backs, wide receivers, and holders for place kicks all throw passes from time to time.
From a playing rules standpoint, there are certain distinctions to be made. For example, in pro football, a T-formation quarterback who takes a position directly under center is an ineligible receiver. In this case, however, the relevant consideration is the player’s position at the time of the snap, not whether he normally plays quarterback or punter. So again, the question is pretty much meaningless.
How about for fantasy football points. I was in a league last year that you got team quarterback point (not an individual). If the punter threw a touchdown, would a person get the fantasy points for team quarterback points?
Oh goody, football quarterback questions.
So, who’s the quarterback on the defending team? Who can pass (and receive) after an interception?
Nobody, in none of the above, if you’re an Oakland Raider. 
Peace,
mangeorge
Whoever is passing the ball, be it the QB, another back, a punter on a fake, it’s important to remember that the penalty is “roughing the passer”, not “roughing the quarterback.”
So no hitting the guy after he throws it, OK?