As far as I know (correct me if I'm wrong), SpaceX have only reused previously-wet farings on their own Starlink launches. They don't risk potentially damaged parts on external customers.
Yes.
They haven't been very public about how much refurbishment is required for fairings that they fish out of the ocean, but they've managed to use them on Starlink launches.
There are a lot of electronics on these things and weight is ultimate factor.
Notice that with RocketLab reuse, the helicopter is staged from a ship at sea. However, probably can save more money by recovering a RocketLab Electron first stage than half of a fairing (you’d need one helicopter per fairing halve but only one helicopter for Electron... Electron also has a much smaller and less awkward aerodynamic cross section than a F9 fairing halve). See: https://m.youtube.com/watch?v=enndCzvZpZk
The problem with this and a fairing is that if the fairing catches the open side, it's going to drag behind like a giant parachute and take the plane down. The fairing is huge (13m long x 5m diameter), even compared to a big plane.