Let's say we have a 50mm lens on a full-frame reflex camera. As mirrorless cameras have a smaller distance to the sensor, does it mean we need a shorter focal length for the same sensor size to have the same field of view?
(Related What is a normal lens?)
No, we don't need a different focal length – a lens designed for a mirrorless camera is just intrinsically different to a lens designed for a traditional (D)SLR camera. The image that a mirrorless system lens projects out the back and onto the sensor is designed for the shorter flange focal distance.
I feel like this schematic (taken from DPReview's very old Mirrorless camera buying guide) shows it quite clearly. You can see how the lenses are designed to project the image differently for the respective flange focal distance.
That's why/how you can mount a traditional (D)SLR lens on a mirrorless camera – all you need to do is mount it further from the sensor (with the help of a spacing adapter).
No. The actual length of the lens does not correlate with its focal length, at least not directly.
The rear element of a typical 50 mm lens on a full-frame mirrorless camera is a bit inside the lens barrel (i.e., some distance forward, into the barrel), roughly the same distance from the sensor or film plane as the rear element of a typical 50 mm lens on a reflex camera, given a similar optical design.
This changes for wider angle lenses (shorter focal lengths), that can have their rear elements physically placed closer to the film plane without any special optics tricks. Whereas wide angle lenses on reflex cameras, say 20–24 mm, begin requiring retrofocus groups in order to effectively shorten the focal length of the lens while simultaneously keeping the rear element far enough away from the sensor or film plane so the reflex mirror can swing up.
No, it means the lens' rear nodal point must be that much further in front of the flange so it can be the same distance in front of the sensor.
Let's assume we have a Canon EF lens and a Canon EF-M lens with the same lens formula. Each are the same focal length.
Let's say the rear element of the EF lens is even with the flange ring. That means the rear element is 44mm in front of the sensor. The same optical formula with an EF-M mount would need to be 26 mm in front of the EF-M flange ring. The 18mm between the camera's flange ring and the sensor is added to the 26mm between the lens' flange ring and the rear element. Thus in both cases the rear of the lens is the same 44mm distance from the sensor, it's just that some of the distance has been shifted from the camera to the lens, and an equal amount taken away from the camera.
If we want to use the EF lens with the EF-M camera, we must use an EF to EF-M adapter that is, you guessed it, 26mm thick so that the EF lens' flange ring is 26mm in front of the camera's flange ring that is 18mm in front of the sensor. 26 + 18 = 44. So the lens' flange ring is now 44mm in front of the sensor, as it was designed to be.
Where it may get confusing for some is that different lenses with the same focal length but having different optical formulae may not have the rear elements the same distance from the sensor. That doesn't mean the rear nodal point isn't in the same place. The rear nodal point is where a single element lens with zero thickness would be located to give the same focal length as a lens with multiple optical elements.
You need a different flange distance, not a different focal length. Flange distance is sometimes referred to as "flange focal length" or "flange focal distance"... but that is best understood as the flange distance portion of the focal length/distance.
Focal length is the distance from the second principal plane (nodal/primary point) to the second focal point (image plane). This drawing shows the principal points in a simple (thick) lens.
This shows the second principal plane of a more complex 28mm lens design (labeled as point of convergence).
Here I have added an element and second principal planes to the drawings @ossulic provided to make them correlate more accurately. It shows the focal length as 50mm for both, but with different flange distances (44mm/18mm) and different placements of the second principal plane in front of the lens mount/flange (6mm/32mm).
Note that the drawings are not to scale nor a correct combination of elements... depending on the lens design the second principal plane can fall anywhere inside a complex lens; or even outside of the lens itself (e.g. a telephoto lens).
And finally, a lens' focal length defines its' angular field of view; it is not variable. I.e. all true 50mm lenses have the same angular field of view.
$$\mathrm{AFOV} = 2\tan^{-1}\left({H\over2f}\right)$$
However, different sized sensors will record different amounts of that field of view... so if your mirrorless camera also has a smaller sensor ("crop sensor"), then you would need a different focal length for the same recorded field of view.
