How do a helicopter's cyclic and collective transfer commands to the swash plate?
The mixer is mainly found in larger helicopters where their expense is not such a large part of the purchase price. This answer explains more of the function of the flight control mixer: it de-couples degrees of freedom from the control inputs.

The helicopter in OP picture shows unmixed inputs into the swash plate: tilt swashplate forward for longitudinal control, sideways for lateral control, pedals connected to tail rotor only. Picture below is from Principles of Helicopter Aerodynamics by Leishman, and shows a schematic of the cyclic longitudinal

Three actuators are connected to the Non-Rotating Plate, which can be tilted and lifted. The Rotating Plate follows the fixed plate, and cyclically pitches the rotor blades so that they adjust their circular path up/down. The swash plate arrangement is quite complicated, as shown in the photo below, from the same book.



The mechanical linkage from the cyclic and collective sticks to the main rotor actuators does have some input summation involved, as the picture below shows (from this source, which indeed calls it a mixer).



There are always three swashplate actuators. The collective stick always deflects all three simultaneously so that the swashplate lifts, while cyclic stick movement deflects one or two in such a way that the swashplate tilts.