I'll assume you are at the 500mm end of the lens, since your questions seem to be those of someone trying to get extra reach to shoot little birdies and wanting to retain as much detail as possible. (I'm often in the same boat.)
#1, *unless* you are working with extreme dynamic range situations (not necessarily super high ISO's) where you will need to lift shadows in post and the D7100's banding issues in deep shadows become a problem. If you're after more reach, go DX.
#2 involves using optics to spread the image from the 200-500 out over a wider area. (That's what teleconverters do.) Instead of a smaller image illuminating 24 million small pixels, you're using optics to spread that image out to illuminate 24 million big pixels instead. If your lens and teleconverter are both optically perfect, you'll wind up with basically the same picture either way. There's no point in doing this: you'll have about the same total amount of light hitting the sensor, and the increase in image quality inherent in the big FX pixels will be canceled by the higher ISO you'll need to use, since you're now at f/8. This assumes your optics are perfect, though; real teleconverters cause aberrations on their own (independent of magnifying aberrations caused by the lens, which gets canceled by the bigger FX pixels). The TC-14 is good but it's not perfect.
#3 is objectively worse than #1: Suppose you shoot an image on a D750, and then crop it by 1.5x on each side to match the "level of zoom" that you'd get from the D7100. You now have the light from a DX-sized patch on your sensor spread over 11 million pixels. Wouldn't you rather spread that over 24 million pixels and get more detail? That's what the D7100 does for you.