Lens: Zuiko OM 24mm F2, Camera: Sony A7S2
What would happen if a focal length reducer (0.71x) is used to adapt a 24mm F2 FF lens, into a FF camera? The resulting field of view should be ~17mm with vignetting, which after applying an in-camera 2X crop would become 35mm clean. But what about focal length and depth of field? After the 2X crop, would it become an equivalent to a 35mm F2, or a 35mm F1.4? Does this sound like a good idea anyways?
TL;DR: Use the 24mm lens directly. Forget about the focal length reducer.
(If the lens is compatible with your body, and if such a focal reducer exists - which I don't know. If they are incompatible, better get a compatible lens than go through the hassle of using any kind of adapter.)
You're using some terms that need clarification:
Field of view (FOV) should ideally be given in degrees, but alas we are all familiar with the field of view that a given focal length produces on full-frame sensors (or 35mm film). So, we often find field of view expressed in millimeters. So, a "35mm field of view" effectively means 63 degrees diagonal. You use that interpretation in "field of view should be ~17mm with vignetting".
Focal length is a property of an optical system (lens or lens combination) measured in millimeters. Often it is mis-used as a way to communicate field of view, assuming a full-frame sensor. "applying an in-camera 2X crop would become 35mm clean" is such a mis-use. Obviously cropping in the sensor does not change anything about the optical system, it only reduces the field of view by deliberately ignoring some part of the image. So, a more exact wording would be: "applying an in-camera 2X crop would give a field of view equal to a 35mm lens on a full-frame sensor."
Depth of field (DOF) measures how much depth in the object space (in front of your lens) you get "sharp" in your image. That needs a definition of the word "sharp". Any object that is not exactly focussed will produce a "circle of confusion" instead of a sharp point on the sensor, and in the old 35mm times, a circle of 30µm diameter was used as the limit. But what matters for the sharpness impression of an image is not the absolute size of the circle of confusion, but its size relative to the whole image, so we'd better express it e.g. as 1/1400 of the image diagonal.
Depth of field for subject separation (added based on a comment of OP): We often want the subject separated from the background, meaning that we want fine details of the background to be smoothed out. Then the question is, how far beyond the focussed object e.g a 1cm object gets "blurred out". You can imagine the rays from an object point to the lens to form a cone that reaches the lens opening diameter at the front of the lens, and a mirrored cone beyond the object. In other words, the question is, where this mirrored cone reaches the 1cm diameter. For a given distance to the object, that only depends on the lens opening diameter. E.g. for a 1m distance and a 20mm lens opening, this will happen 0.50m beyond the object.
Speed/aperture: This can be understood (in a simplified model) as the ratio of the lens opening to the focal length. F/2.0 means that the effective lens opening diameter is half the focal length. All the light that an object (inside FOV) radiates into this lens opening will hit the sensor.
To your questions:
The resulting field of view should be ~17mm with vignetting
Yes, true (denoting the FOV angle with the full-frame-equivalent focal length). And here, the 17mm value also gives the true new focal length. Especially the vignetting will probably be so heavy that the corners might be completely black. So the term "field of view" sounds misleading if you can't see anything in the corners.
which after applying an in-camera 2X crop would become 35mm clean
Yes, if you apply a 2X crop, you get half the FOV. And if you express that angle as full-frame-equivalent focal length, the 35mm are correct.
But what about focal length?
The focal length is a property of the optical system, and not affected by cropping, so it is still 17mm.
But what about [...] depth of field?
For a useful definition of DOF, the circle of confusion should be taken relative to the image size, so cropping by 2X should also adjust the circle-of-confusion limit to half its original size, so effectively that cropping also reduces the DOF by 2X. If you crop an image by 2X, you'll probably later enlarge it twice as much as the original one, thus also enlarging the confusion circles twice as much, making them much more noticable.
would it become an equivalent to a 35mm F2, or a 35mm F1.4?
In an ideal world (if the reducer transmits all the light without introducing vignetting itself), the combination becomes a 17mm F/1.4. The lens opening stays the same in millimeters, being 24mm/2 = 12mm. Still we assume that all the light captured in that lens opening still reaches the sensor (and doesn't get cut off somewhere in the reducer). But now we relate it to a shorter focal length (17mm), giving F/1.4.
You can also understand it this way: the same amount of light is now concentrated on a smaller part of the sensor, so the result will be brighter there.
So, what you get is 17mm F/1.4 lens where only the center part is usable. Don't confuse cropping with optical characteristics, so even if you crop it to a 35mm-equivalent field of view, it still is a 17mm optics.
Is it useful?
No. Use the 24mm lens directly.
The reducer introduces its own optical errors, so image quality deteriorates.
The higher speed F/1.4 instead of F/2.0 sounds good. But if you use the 24mm lens directly, you can simply increase the ISO value and probably get better results. Even if this higher ISO introduces a bit more noise, you have a higher pixel count available for noise reduction.
Using the 24mm lens directly, you have a higher pixel resolution, as you use the full sensor and not only half of it.
For subject separation, the focal-length reducer changes nothing.
