Tyvek is installed on the OUTSIDE of the sheathing. You would not see it from the inside of the wall. I don't understand caulking anything you would see behind the shower wall. If it is an outside wall, I would insulate with the plastic-cased batts, and possibly put a 2 to 4 mill vapor barrier over that.
You do not need greenboard or backer board behind a tub surround.
Sterling(Kohler) has the Vikrell spec sheets on their web site. Check with them on the weight limits.
A cast iron tub is certainly difficult to maneuver, but it is the " top of the line" and if you want it, a reputable plumbing contractor would do it. Labor cost would be appropriate to the job.