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It HAS an Auto Focus button. Works great! As for the Anti-vibration/shake, NO, not that I can see. However, a hand stabilizer handle should help with that. I haven't notice any vibrations although I'm still experimenting. Beautiful video! FYI,Can't find an external monitor that works with this camera. Purchased two different monitors and five different HDMI Cables, No luck. But the flip out built-in is fine. Good quality, I just wanted something larger. 

Yes, it works with any camera as long as the camera has a external mic input. But you need to set the output volume to low levels, and listen on the headphones jack of the A7s to make sure you don't overmodulate the audio into the camera. The Sony takes only mic level signal. 

No. This is manual adapter. I've used it for a vintage Takumar 70-200mm and it worked well if all you want is to mount it and have manual control. 

Yes, the tripod can fit your FujiFilm X system camera