The 70-200 Mark II combined with Extender 2X Mark III is worlds better than the earlier lens and 2X Extender combination. Go look in the Lens Photo Sample Archive to see if you can find some examples. I've see images posted here that were great, that I'd be quite happy with. I will not use my 70-200 Mark I with 2X Mark II... there's just too much loss of image quality. (I will use it with 1.4X, tho I rarely do since I have two 300mm lenses.)
The 70-200 Mark II and 2X III combo is competitive with the 100-400.
Will it be tack sharp for moon shots? Probably not. 400mm (the 200mm + 2X) just isn't enough reach. My best moon shots have been with a 500mm and 2X. You'll always get better results using a prime lens for such an extreme shot.
Wildlife? Well, it depends. Often focus speed is important, too. I don't know how quickly the 70-200 II and 2X III focuses. The TC does slow down focus to some extent. I use 300/4 and 300/2.8 (Mark I) with 1.4X II and both of those are quite quick focusing. The 2X on 300/2.8 is fast enough, but definitely not as fast as the lens + 1.4X. I don't use the 2X on the f4 lens or 70-200, so can't really compare.
By the time you are spending what the 70-200 Mark II and 2X Mark III cost, you might also want to take a look at the Sigma 120-300/2.8 OS for use with either or both 1.4X and 2X TCs. Not sure which TCs would be the best choice for use with that lens. Of course, it's a much bigger and heavier lens so you'll likely want to use a tripod or at least a monopod a lot of the time.
If your budget is big enough for the Sigma 120-300 or the 70-200... You also might want to consider the SIgma 120-400 OS or 150-500 OS. Both of these are pretty comparable to the 100-400 (they tend to be a bit less contrasty and deliver a little less saturation, but both of those can usually be fixed pretty easily in post production). Both these Sigmas are considerably less expensive than the 100-400, plus are two ring zoom designs, which personally I prefer over the push/pull single ring design of the 100-400.
Still waiting on the 200-400/4 IS 1.4X Extender... and hoping for a big lottery win so I can afford it!