No it does not.
You can use the controller while it plugged into the PS4 with its USB cord so I guess that would technically make it wired.
I have no personal experience, but it should theoretically work with most bluetooth connections by holding the share and PS button. Once the light bar begins flashing, look for new bluetooth devices on whatever you want to use it with. My wife uses this model controller with her iphone.
Yes I believe it is. The controller that originally came with the PS4 at launch did not have the light in the touch pad. Gen 2 does