Yes, though this is not usually seen often due to the punt receiver's focus being only on catching the ball.
Only if there was a penalty against the team that blocked the punt, such as offsides.
No. Statistically, the attempt and the negative yardage from a blocked punt should be assessed to the "Team," not the punter.
A 'muff' is generally used with punting and occurs when the player that is catching the punt drops it or 'muffs it' without ever having possession of the ball. There is a distinct difference in the NFL rules between a muff and a fumble. If a punt receiver fumbles a punt, that means he had possession of the ball and then fumbled. The punting team can recover a fumble and advance it as many yards as they can get. If a punt receiver muffs a punt, that means the receiver did not have possession of the ball. The punting team may recover a muff but may not advance the ball. The punting team would gain possession of the ball at the spot of the recovery.
Yes they can, although that is almost impossible.
It's a touchdown.