The controllers used for this video are probably the index 'knuckles' controllers that have some finger tracking ability. Therefore good replicas of your hand can be generated given the data passed about the location and rotation of the controller and the general position of each finger. Arms are quite a bit harder as the shoulder joint and wrist joint are both muliaxis (while the elbow is fairly fixed). While the location of the shoulder can be inferred based on the location of the visor and the relative location of the controller the full IK chain can solve a number of ways (With your hands on the keyboard notice that you can change the position of your elbow without much effecting your hand or shoulder position.) because of this an improper solve can throw off immersion, better and easier just to have 'phantom hands'.

Different games will do things differently. Some will display hands, others will display the controllers. You can do either.

After some time, you forget you are even holding controllers (on the Rift, I'm not sure about those clunky Vive controllers, and I haven't tried the Index).

Probably a mix of immersion and practical reasons. It looks like Valve is supporting a huge variety of input devices. Some, like the Index controller do track individual fingers. In a grittier game like this it would be weird to see the Vive magic wand floating around in space. With the gloves, they can also incorporate things like a health bar that would normally go on the HUD.

Yes, it does work great in general, not just in VR. It just that IK can't know if you have your elbow to the left or right, which would be confusing.