HDMI output can only play video and photos. The connection can be connected in any mode of the camera. 

Yes, the video camera can record while it is charging. 

Yes, it can work as a webcam as long as your device can connect to the camera. 

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.