You will need to cut length-wise the insulation to less than 15" wide. If you really want to staple the paper to the stud, cut the insulation the width of the opening plus what you need for paper and then cut about an inch of insulation off the entire width but, do not cut thru the paper thus creating the paper edge needed to staple to the stud.
Yes, I would install a moisture barrier all around the entire room.
Yes, I would install insulation behind the tub surround.
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