2 is irrational
square root of 2 is 1.4142136 etc it is irrational to think that that this keeps it from being usefull

1) 100 lines/mm
Back in the old days of 35 mm Film
Make one or more grids of equal width black and white lines
Take picture
examine negative under microscope
Find the case where the white line is 50 % the density of the black one (or visa versa since we are using negative film)
Count the number of lines/mm at which the 50% criteria is achieved- for example 100 lines /mm

Very similar conceptually to the technique DXO uses in their widely accepted testing

2) Motion Blur
Accepting that motion blur is only one of many factors affecting picture quality

Actually blur resulting from hand holding can be evaluated quite easily using the "sharpen/shake reduction" filter found in the latest PhotoshopCC
A graphic is provided that shows the amount and direction of any motion blur
Experimentally I find that on a D800 with a 400 mm lens with latest VR on, at 1/1000 second , that with care, hand held is indistinguishable from tripod mounted

Different cameras/lenses may produce different results

Fortunately the improved hi ISO performance of modern cameras allows decent pictures in reasonable daylight conditions with these settings (IMHO)
Actually I carry a monopod in most cases

TIFF Files are bigger since they are in a 16 bit format even tho most of these bits are never used they still take up space
Raw files are 10 or 12 or 14 bits per pixel depending on the camera and settings- A good camera can capture about 14 bits
Raw files do not necessarily "have all the data" my D800 has options for amount of compression of Raw --lossy vs lossless etc - The book says loss of quality is negligible
Standard JPEG is 8 bits per pixel with various amount of lossy compression

OK
Since no one else seems to have directly answered the question!

300 mm is 300 mm
The central portion of your shot is EXACTLY the same whether shot in FX or DX
In DX mode the files are smaller which might be important
in FX mode you get more FOV which might be important

RE TC Your instincts are correct, with their use pictures are darker. autofocus may not work and unless you have a very large high quality sharp lens and a camera with limited pixels the resulting shot will be effectively "cropped in Camera" and magnified. with no improvement in quality.
If this feature is useful to you, as it seems too be for some on UHH, then the TC is for you. (or if you have one of the aforementioned lens/camera combinations)

Yea!!
Someone finally answered the original question
I like this explaination

Hey this is a great question! I learned something about JPEG modes
The following is based on my D800 manual

1) Note the difference between quality (amount of JPEG Compression) and size (number of pixels)
2) In both cases it only applies if you want the camera to create a JPEG. RAW files are not affected
3) note that the different "sizes" are in an exact ratio of 3:2
So it is obvious that for example to create a "M" from a "L" one takes 9 pixels (row one column 1, row one column 2, row 1 col 3, row 2 column 1, row 2 column 2, row 3 col 3) and interpolates them, average or median or some other function to create a new pixel row 1 column 1 of the smaller size file

Since each new pixel interpolates 9 old pixels the noise will be significantly reduced (very complicated math here)
Note that no pixels are "thrown away" they are interpolated to create new pixels
3) Crop is something else again, in crop mode pixels not in the defined saved central region are thrown away

Note that for other ratios 2:1 for example as for your camera a similar algorithm applies, this works conveniently for simple whole numbers like 3:2 , 2:1 , 4:1

OOPS it is not clear how to handle the discontinuity between row 4 and 5 but it probably is not visible to the naked eye

Yes the post office does have rules . As I recollect it involves the number of batteries

Yes airlines (DHS) does have rules on lithium batteries, go to DHS web site. There is a limit on the number of batteries but batteries in the camera or in a battery grip are OK, but a dozen "spares" would not be OK

Hi Peggy
Assuming the lens we are talking about is the EF 70-200mm f/2.8L IS II USM (it was not clear from your post)
wow that is a great lens, then following the thoughts of previous posters, it is probable that some net improved resolution could be achieved by use of a (or more than 1) TE

Did the auto focus still work with that stack?

The gist of this thread as I understand it is to determine which combinations of lenses and camera will benefit with improved resolution (not just magnification) by use of a TE

By chance did you compare shots of the same scene with the TE and without TE (digitally zoomed in computer to provide the same FOV. You mention "some degradation" but it was not clear what that was relative to
Thanks

It is good to see your post with facts rather than just opinion

Let us return to the original question which was (to paraphrase)
"will adding a TC to a given lens improve the ability to resolve small items such as BIF"

it was NOT

"will a long lens with TC provide better ability to resolve small items such as BIF compared to using a different short lens without TC, when said pictures are cropped to equivalent FOV in PP"

That someone would propose such a comparison speaks volumes

I completely agree with this well stated post.
A TC might be of use if you have a new expensive lens and an low pixel count camera body, otherwise it will make things worse
Anyone who puts "fill the frame in the viewer with no need to crop post processing" as a higher priority than "allows smaller aperture for better FOV" or "allows faster shutter speed for less shake" or "allows lower ISO setting for more dynamic range" is not any bird photographer that I have ever met

The article nor most of the posts in this thread did not address the question , which was "did the use of TC provide a better (higher resolution) picture than that which could be achieved by simply cropping the original image in PS"

Some have suggested "it all depends" which I think is the correct answer- if your camera has enough usable pixels to display the smallest element resolvable by the lens (IE more pixels would not produce a better picture) then it is possible that a TC might produce a higher resolution picture, dependent on a host of other factors

Why do you ask?
It seems that I have used my left eye for years for no apparent reason other than habit.
One might suspect that right eye is better since it centers the lens on the head/body better providing better stability

I second the thoughts of Billyduds
The D800 and 70/200 are well matched as is, so it is unlikely that the TC will do anything other than reduce the FOV, increase noise and slow down focusing. With a bad camera and that good lens the TC could help- you have 1/2 of that mix.
Concur on the BBF. I have the D800 w 80/400G, combo is a bit heavy but gets good bird pics

Rather than more or fewer bees, how about big bees vs smaller bees

The big bees gather more light but they create a rougher textured picture (IE noisier)

Love my D800
One thing I had to learn was how to make effective use of those extra pixels
Nikon suggests to use a tripod!
I frequently find it useful to shoot higher ISO than I did previously as the improvement in shake blur overcomes the higher ISO noise much of the time (all things in moderation)
Even with the VR I find the old formula of shutter speed=1/(2 X FL) to be valid
Even if you do not use the Shake blur filter in PSCC the analysis graphic that comes with it will tell you how you are doing on shake