Answers

Probably need to have the skylight reflashed.  Skylights need to have special flashing kits designed for skylights.  Have a professional come to inspect to see if you have a curb mount or deck mount skylight and they will be able to give you a price to replace your current flashing with a new airtight skylight flashing kit.

First make sure there is weather stripping between the skylight frame and the curb, assuming this is a curb mount. sometimes there is not a seal inserted at that joint. If that is not an issue, then you could certainly install a frame around the well at the ceiling level and insert a piece of clear acrylic to keep cold from dropping into the room. I have done this at a previous residence with great success. Good luck

Without knowing the specifics of the skylight brand and condition, we would not be able to answer the question with confidence.

We use Velux skylights. During re-roofs, we typically find most skylights are not in good condition and/or were not installed properly.