Down thrust is typically neccessary on high wing cabin type or parasol airplanes. On these aircraft where the thrustline is located well below the center of mass of the airplane, thrust will tend to pitch the nose upward and every time the power is changed there will be substancial pitch change and down thrust tends to minimise this. Mid wing or low wing airplanes usually never need downthrust.

The purpose of the right thrust is a combination of factors with 'P' factor being the most important. This occurs strongest at high deck angles, low speed and high power settings, It always results in a yaw to the left around the vertical axis. This is because the right side of the propellor disc is producing more thrust under the condidtions mentioned earlier. These descending blades are at a higher angle of attack than the ascending blades. This condition is minimised not entirely eliminated by the use of right thrust. To answer your second question, Yes rudder is the proper corrective action to prevent this yawing. The use of aileron to correct will actually make things worse. However if you trim the rudder to correct this which only occurs at takeoll and climbout or high angles. Then for the rest of the flight you will be flying around in a right skid, not real purdy. This is the primary reason why right thrust is used on many airplanes.