I use the 1.4X Mark II on 70-200/2.8 IS Mark II and it's a good combo. I have heard the Kenko Teleplus Pro 300 DG or DGX 1.4X is equally good and a bit cheaper (note: supposedly the newer DGX has been improved to provide faster focusing, or at least less loss of focus speed). Some have reported good results with the Sigma and Tamron TCs, too. I would avoid cheaper TCs, such as the Kenko MC4 1.4X and 2X. They are less expensive, but don't seem to offer nearly as good IQ. I don't know if the Kenko Pro 300 2X is worth trying or not.
I tried using the 2X Mark II with my 70-200, but did not like the results with it. So I won't use it (of course it helps that I have 300/2.8, 300/4 and 500/4 lenses as an alternative). I reserve the 2XII for use on the long primes, where it works very well.
The 2X Mark III appears to work pretty well, judging from the sample above. I haven't any experience with it myself.
You can fool the camera into focusing... or at least trying to focus... when using the 2X with an f4 lens. Most of the cameras are programmed to turn off AF when the combo gives an aperture less than f5.6 (f8, center point only, on 1-series cameras and the EOS-3 film camera... except for the new 1DX which apparently is going to be limited to f5.6 too).
Just put some tape on a couple of the contacts on the back of the teleconverter, and the camera won't know it's there and will try to focus. It might be a bit slow, hunt and fail in low light, but it might work adequately for you. With really good light and contrasty subjects, I can get my 7Ds to focus - with limited success - at effective f11 aperture. They actually do pretty well at f8. Haven't tried 5DII yet... but it should be better (the 5DII's AF system, though more primitive in many ways than the 7D's, is rated to work in about 1 EV lower light than the 7D's). If you use this trick to fool the camera, the EXIF of your images will have some incorrect metadata... it will show lens focal length and max aperture as if the TC weren't there. But, AF and metering/exposure all will work just fine.
If interested in giving it a try, search here or elsewhere on the Internet for more info about which contacts to tape up on the teleconverter.