I have a Canon EOS 7D Mark II (1.6x crop factor), and I have a Canon EF 100-400 IS USM. Working with just 400mm for the purposes of this question, my understanding is that my 35mm-equivalent focal length is 640mm (400 X 1.6 = 640). Now, if I purchase a Canon Extender EF 1.4X III, does it take my 35mm-equivalent focal length to 896 (400 x 1.6 x 1.4 = 896)? It seems like I'm missing something!
Well, the main thing is that crop factor doesn't really affect focal length. It just affects the field of view by making it narrower. So, what you really have is a 400x1.4x => 560mm lens combination on a crop body, which has the same FoV that an 896mm lens would have on a full frame body. So, unless you shoot full frame enough to translate focal lengths to given fields of view, the 896mm figure may be kinda meaningless to you.
See the Canon Field of View Comparator tool to get a sense of what crop factor on a given lens means.
You also should be aware that the 1.4x teleconverter (TC)+ 100-400L combination, while it will work on a 7DMkII, wouldn't work on a lot of other Canon crop bodies. Using a TC is not exactly the same as using a longer lens. The TC, while it increases focal length, by the very act of doing so, reduces the lens's maximum aperture. A 1.4x TC reduces it by one stop, so once you put it on, the 100-400@400mm effectively becomes an f/8 lens. And most Canon crop bodies can't autofocus at apertures that small--they mostly stop autofocusing once the max. aperture of the lens hits f/8.
Yes, placing a teleconverter on a crop works exactly the same as on a full frame – you multiply the teleconversion factor by the lens focal length, and since it's on a crop, you factor in the 1.6x crop factor of a Canon APS-c to get an effective focal length of 896.
Keep in mind that placing extenders on any lens will decrease its effective aperture. For the 100-400 which is already fairly slow, this could bring the aperture down past f/8, which I seem to recall is the minimum allowed for the central focus points on a 7dII.
Yes many people do exactly what you suggest and end up with a field of view that is equivelent to an 896mm lens on a full frame camera.