aerodynamicsaircraftclassical-mechanicsfluid dynamicsnewtonian-mechanics
In most helicopters there is the anti-torque tail rotor to prevent the body from spinning in the opposite direction to the main rotor.
What's the equivalent mechanism in horizontal takeoff single engine propeller, and jet aircrafts, where the air or the jet coming out back from the turbine or propeller is spinning and will cause such aircraft to roll?
The air is of course the key to understand your questions. To start with the last question, as you figured out right yourself, in vacuum body of the helicopter would not turn at all. But in vacuum helicopter could not fly at all!
At least part of the principle of rotor is that by pushing air down makes helicopter airborne. However in order to do that, blades of the rotor are a bit inclined, so when blade hits air right with velocity $\vec{v}$, it pushes air mostly down and but also to right with force $\vec{F}_\text{B}$. Third Newton law tells us that in that case air pushes blade mostly up, but also to left with force $\vec{F}_\text{A}$.
The vertical push $\vec{F}_\text{V}$ makes helicopter airborne. However, the horizontal push $\vec{F}_\text{H}$ to blades creates torque which changes total angular momentum of the helicopter. In order to stop body of helicopter spinning you have to create oposite torque and you do that by tail rotor (or in your case by friction of the floor). If there is no tail rotor or friction of the floor, body of the helicopter would spin faster and faster until the drag of the air due to body of the helicopter spinning is so large, that creates apropriate torque.
My apologies, I won't be reading your entire question.
But still I will provide an answer. Why is that? Because flight does not require any of the things you talk about.
You could build an airplane that would fly with no "airfoil" shapes. You could build an airplane that would fly with completely flat rectangular wings made out of plywood. The important thing would be the angle of attack of the wings to the air. Consider a flat piece of wood, like plywood. Push it through the air in a direction exactly parallel to its flat dimensions and it develops no lift. Tilt the wood so the leading edge is "up" compared to the direction it is moving and you can feel the lift.
The lift can be thought of a few ways. Think of the air molecules hitting the surface of the wood. They bounce off, in a downward direction. Well if we are pushing air downward, we must have an equal and opposite force, which is the lift. Or another way: we are gathering air under the board, it gets a little pressurized. The pressure is pushing up on the wood. This is really the same picture as the first if you think about it.
All the rest with airfoils and so on, all this has to do with developing lift efficiently, developing lift while minimizing drag. An airplane with flat plywood wings would fly, but it would have a lot of drag and would therefore be very inefficient.
Best Answer
The motor does exert torque on the fuselage.
The pilot, without having to think about it, compensates by applying right aileron, which has plenty of roll authority.
There's more to it. When a propeller-driven plane is taking off, it has a tendency to yaw to the left, and the pilot automatically applies right rudder to compensate. That left-turning tendency is due to the propeller descending on the right hand side, at a higher angle of attack (thus more thrust) due to the pitch of the airframe, and also because the wind coming off the propeller is a corkscrew flow, and it strikes the left side of the vertical stabilizer.
If you watch this video, you will see how pilots are instructed to operate the F4U Corsair in WW2. It is recommended to tune in some right-rudder and right-aileron trim when taking off. It is also recommended not to take off at too slow a speed because it will seem "left-wing-heavy" due to that torque, and more speed means more control authority.