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. 

No, you can use the recorder directly. It has a clear OLED display and is very easy to use and set up. The recorder with playback function also supports connection to computer (Windows/Mac) and cell phone (Android/iPhone), so you can manage or transfer files easily. 

No, you need to buy it separately, which is also available on K&F Concept. 

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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.