I shoot FX and DX. I have a little different perspective on this.
First, my use of "angle of view" and "Field of view" are identical. Perhaps you might want to elaborate on your usage of those terms.
If I shoot a scene on a tripod at 12mm DX and 18mm FX I will get *essentially exactly* the same perspective and object sizes, except for minor differences in the lenses. It is very difficult to get two lenses at the precise respective focal lengths. And even if the tripod is not moved, slight differences in camera size could result in very minor differences.
This sort of comparison is typically done with two different lenses, or the same lens zoomed to try to approximate the intended focal lengths. There are variances in labeled focal lengths and reported EXIF focal lengths.
There are also variances in the pincushion or barrel distortion of the lenses, or the same lens at radically different focal lengths (such as 12mm/18mm). But these are not fundamental FX/DX differences. They vary by lens and each lens stands on its own in this regard.
My D300 has 100% viewfinder coverage, but my D700 only 95%. That alone makes it almost impossible to shoot the same scene in a precisely comparable manner.
The relationship between near and far objects (perspective) is solely based on working distance, not focal length. Theory agrees with my own observations. I've gone into all this to try to account for the differences you suggest.
As far as "edge distortion", again this is a lens specific issue. Any given FX lens, at 18mm, might have more or less distortion than any given DX lens at 12mm. If that were a primary concern it would require careful study and selection of lenses before deciding on a format.
I agree the 12mm lens will have greater depth of field (DOF). But as a landscape photographer you should consider that an advantage for DX, not a disadvantage.
In the case where the absolute minimum DOF is desired, then FX has a one stop advantage because it "loses" almost exactly a full stop of DOF when the scenes are equalized for perspective and FOV. And that is the only fair comparison.
For landscape work, a certain amount of DOF is required in order to keep the desired near and far objects in focus. That requires the FX lens to be stopped down one more stop than the DX lens in order to achieve comparable FOV.
That extra stop down halves the shutter speed, assuming ISO is held constant. If you shoot from a tripod this is not a problem but in any hand held shooting this actually tends to equalize DX and FX.
You suggest that the difference between the high ISO performance of the D7100 and D600 is two stops. DXO rates it as about 1 1/3 stops. FX has an inherent advantage of almost exactly one stop over DX, in terms of high ISO performance. For example, if you shoot the D600 in DX mode and then FX mode and then test the noise levels (at a standard output size) it will work out to that one stop.
That one stop advantage only works if you do not care about DOF, or want to minimize DOF, or are basically happy with the DOF you get from your respective lenses wide open.
But as soon as you equalize DX and FX for comparable DOF you lose most or all of the high ISO advantage .
You can also lose a stop or so on FX when using a different lens to achieve the longer 1.5x required focal length since lenses generally get slower wide open as they get longer. Depends on the focal lengths of course, and the selected lens(es).
You can shoot 50mm DX or 85mm FX and get about the same FOV and both are available as f/1.4 prime lenses. But if you want to shoot 85mm DX, then you need about 135mm FX and there you will not find an f/1.4 lens. If you are using a 70-200/2.8 then you are at f/2.8 on both formats and not a problem, but with less "low light performance" than the fast primes.
In your theater example, if you are shooting only one actor then FX has that 1 - 1.3 stop advantage. If, however, you are shooting two actors at different working distances and you need to stop down to get both in focus you have lost most or all of the FX advantage, assuming both cameras are of the same era and same inherent noise levels.
At the resolutions of the current sensors, diffraction is a real issue, especially with landscape shots that require DOF far beyond the typical diffraction limits. In principle FX has an advantage here because the same image resolution (pixel count) can be achieved with larger pixels. However, if you have to stop down to maintain comparable DOF then, again, that advantage is lost!
As I mentioned, I shoot FX, I like FX, but I also recognize that DX and FX are closer - in many real world situations - than most people think. And in my particular case, my D700 is inherently less noisy than my D300. With today's offerings that may not necessarily be true, and at least according to DXO's numbers the differences are only about 1/3 stop.
Interestingly, according to DXO's Dynamic Range numbers -the most important spec for a landscape shooter working at base ISO - the Dynamic Range of the DX D7100 significantly exceeds the FX D3x and D4. It is, however, about a half stop under the D600. Again, illustrating that many of these differences are more camera model specific than format specific.