Rudder is usually used to maintain coordinated flight. When banking to one side to begin a turn, the rudder is usually pushed towards that side. I guess this is still valid in inverted flight: in normal flight, if the pilot bank the plane to the right, it also push the right pedal (to conteract adverse yaw); in inverted flight, if the pilot bank to the right, the plane will turn to the right seen from the ground, i.e. the left seen from the pilot. The pilot should apply left rudder to keep coordinated flight.

If the pilot want to make a barrel roll, what will be the general action on rudder given the flight goes quickly from normal to inverted and back to normal, and the pilot don't want to initiate a turn?

Rudder is usually used to maintain coordinated flight. When banking to one side to begin a turn, the rudder is usually pushed towards that side. I guess this is still valid in inverted flight: in normal flight, if the pilot bank the plane to the right, it also push the right pedal (to conteract adverse yaw); in inverted flight, if the pilot bank to the right, the plane will turn to the right seen from the ground, i.e. the left seen from the pilot. The pilot should apply left rudder to keep coordinated flight.

If the pilot want to make a aileron roll, what will be the general action on rudder given the flight goes quickly from normal to inverted and back to normal, and the pilot don't want to initiate a turn?