Focal length is focal length, a 50mm lens is a 50mm lens, whether it's on a 110 cartridge camera, an APS-C digital, 35mm/Full-Frame, 120 roll film, or a large format sheet film camera.
Its focal length, and the perspectve it offers wont change depending n the size of sensor behind it.
The crop factor is how much of the frame/sensor behind the lens, is captured in the image. So, smaller sensor will 'crop' more off the edges, and capture less of the image circle so it will deliver a smaller angle of view, and enlarged out of the camera, that smaller image area will have the same magnification effect as a lens of longer focal length, but not the same perspective/DoF.
Optimized for use on a smaller format frame/sensor, the lens need not produce such a large image circle to cover it,and cropping more off the edges of that mage circle, edge distortion, vignetting and fringing effects, will inherently be excluded, so the lens need not be manufactured as critically, so can be made more cheaply. Hence disparity of price between lenses of the same focal length made for smaller format cameras to larger ones. & incidental effect that taking only the 'sweet spot' from the center, an FX optimized lens of any focal length, will tend to deliver (argh) 'sharper' results on DX crop body camera, or vica versa, a DX optimized lens, on Full-Frame FX may distort or vignette at the edges, it's not been optimized to provide coverage for.
But, bottom line s focal length is focal length, and 'The Crop-Factor' is merely an 'equivalence' for the field of view they give on different sized sensors.
A 35mm on DX wont give the 'same' mage as a 35mm on FX, nor will it produce the 'same' image as a 52mm (35mm x the 1.5x crop factor) on an FX... will give the same field of view, effective subject framing/magnification, BUT, a 35mm lens is a 35mm lens, and will have perspective & DoF of 35mm lens, regardless of the size of sensor stuck behind it.
Eg: I have a 4.5mm 'full round' fish eye lens for DX.. edges by nature are rather distorted, but with such a short focal length, the Depth of Field is pretty enormous, at pretty moderate apertures, everything fro almost the front element of the lens to infiity is in acceptable focus. I also have a little action-cam, which also has a 4.5mm lens, only that has a micro-sensor behind it, the crop factor is enormous, and gives it the same field of view as a 22mm lens on DX something like a 5x crop factor.. BUT it's still a 4.5mm lens, and the DoF is the same as the fish-eye.... convenient, as it makes the cheap little action cam effectively 'focus free', saves a focus mechanism, and any focus lag for video.... and it chops all the fishiness off the edge of the frame.
Just for note; that DX optimized 4.5 fish could be used on an FX body.. already delivering the full, un-cropped full-roond image circle of the lens, though, on the larger sensor body, you would get exactly the same image projected onto the sensor, and it would be exactly the same size.. it would just cover less of the sensor area, so when viewed at the same viewing enlargement, the circle would be smaller, ad there would be more masked area around the image circle in the frame. Which takes the matter to the limit for illustration of the principle
Crop factor ONLY applies to the effective Field of View or angle of coverage, when you use a lens on different film/sensor formats. Lens remains whatever its focal length happens to be, and all other properties are consistent to that whilst coverage is a function of the film/sensor size it has been optimized for.
