Illegal Touching -- Punts

[Johnny Cakes]

Is it correct that every time the punting team "downs" a punt, it is technically an illegal touching, with the remedy being that the receiving team is awarded possession at the spot of the touching?

Second question: Suppose a punt is rolling, the punting team touches it, but it keeps on rolling and then the receiving team picks up the ball and runs for a touchdown. Is it downed where the punting team touched it or is the touchdown valid.

For both questions, assume that all touching occurs beyond the line of scrimmage.

Thanks.