Most airplanes have a rudder on top of the tail. When it is turned to the left, it causes the plane to yaw to the left. But it also puts a roll force to the right, because it is usually above the center of gravity. Of course, this force is smaller, as the leverage is much shorter. Still, this is not logical to me. Wouldn't you want a roll force in the same direction as the yaw? Or at least no roll force at all? It would be more logical to me if the rudder was below the center of gravity. Why isn't this usually the case? Is there a name for a rudder below the tail?
It is bad to put things below the tail because they might hit the ground. Even with nothing below the tail sometimes there is still a tail strike.
The MQ-1 Predator does have a V-tail ruddervator below the tail.
The MQ-1 is already at high risk of a rear propstrike. As a result, there is not much disadvantage to inverting the ruddervator.
More on V-tails: Why are military drones shaped so strangely?
Why isn't the rudder below the tail?
Compromises, as usual in the aerospace world.
From an aerodynamic and flight mechanics point of view the best location for the vertical tailplane would be aligned with the centerline of the fuselage, i.e. sticking out both upwards and downward. This would give a pure yaw moment with the deployment of the rudder without any cross coupling with the roll movement, as you have correctly highlighted.
Something similar has been seen in the past, like for example the Douglas XB-42:
In that case the lower part of the vertical stabiliser had the additional work of protecting the propeller from striking the ground. This kind of tailplane is normally termed cruciform.
Ok, but what about taking off? When the takeoff speed is reached, the AoA of the wing has to be increased to get enough lift to takeoff. This manoeuvre is called rotation and it makes the bottom of the fuselage almost scratch the ground (A380 image source):
If a vertical tail were there, the rotation would be possible only with very long landing gears: the XB-42 had indeed relatively long landing gear and "careful handling during taxiing, takeoff, and landing was required because of limited ground clearance. An integral shock absorber was added to the ventral fin to reduce excessive bending force in the tailcone from a ground strike".
This solution might work for a small airplane but not for a jetliner: longer landing gear means higher weight, bigger space needed in the wing for retraction (instead of fuel) and higher stairs to board the airplane. Everything is a no-go and so, as a compromise, a vertical tailplane sticking out only upward is used in exchange for some (very small and perfectly controllable) yaw-roll cross coupling.
Note that a similar reasoning applies also to the wing: the best location would be just in the middle of the fuselage i.e. not high like in a cargo, nor low like in any jetliner. But then the wing structure would cut the fuselage from side to side in half, splitting the aisle in a separate forward and rearward section.
Ventral rudders that extend below the vertical C of G have a plus, but a minus.
The plus is rudder application imparts a roll toward the slower wing when yawing, similar to dihedral effect, but which doesn't require sideslip to get started to work. You could say that the rudder's effect as a big aft-mounted aileron is harmonized with its yawing effect in providing a desirable rolling reaction to a yaw input.
On the other hand, sideslip's action on the fin imparts a roll into the slip, counteracting wing dihedral effect (which requires sideslip to be present to do its thing) and increasing its tendency to spiral.
In practical terms, a good dihedral effect is more desirable that a favourable roll-yaw couple, so this favours the rudder above, which also has all the obvious packaging configuration advantages. So under slung rudders are rare.
A couple of cases that come to mind are the Lesher Teal and Molt Taylor's Mini IMP. Molt used to promote the sympathetic roll due to yaw effect as a feature of the inverted V tail.
What we are talking about here is the vertical stabilizer, and yes, it can be below the tail, as seen in a common paper airplane.
But the position of the tail is not the only factor in creating a roll force in the same direction as the yaw.
Let's start out by making the plane stable in roll first. If the plane is tipped to the side by a wind gust, how can we design it to roll itself back to upright?
It's important to realize that once a plane rolls, the wing produces less vertical lift, and the plane will start to sink. This puts more wind flow on the downward facing side.
If the vertical stabilizer area is above the center of gravity then the plane will tend to roll back upright. Wing dihedral also helps do this.
So how do we create a roll force in the direction of the yaw?
By overcoming inherent roll stability with ailerons. This is called "coordinating" the turn.
Designers try to balance between excessive stability and the ability (when desired) to roll, pitch, and yaw the aircraft.
If you just push the rudder, where is the wind flow?
Now it's on the other side, pushing the dihedral into the turn! We can see the dihedral produces a roll torque and the (upright) rudder produces an opposite roll torque.
this helps prevent the need for opposite aileron to "hold" the inside wing up
In design, it's far safer to have the inside (turning) wing aileron up and the outside wing aileron wing down, as this lowers the AoA of the inside wing.
So much for rudder turning, how do birds "do it"?
They pull the inside wing in and slip turn. Here, it is advantageous to have the Vstabilizer/rudder down, because now wing dihedral rolls away from the turn.
So much for the birds, how do most airplanes want to turn?
the best way to turn is coordinated, with the airstream as close to parallel to the direction of flight as possible.
This means that after the rudder is applied in a coordinated manner with banking (rolling) the aircraft, and as the aircraft begins turning, side forces drop to a minimum.
the rudder and vertical stabilizer can be up or down
For practical reasons involving tailstrike, they are usually up. The Junkers Ju-187 was actually designed to have its tail rotate from "upright" to downward in mid-flight.
Finally, there is a device that creates roll and yaw in the same direction known as a spoileron, mounted on the upper surface of the wing. The spoilerons acts to create yaw (by drag) and roll (reduction of lift) on the inside turning wing. Essentially, it is half an aileron.