I can't know if you live in the US or UK, but Duffer has the intial clue.
"Codes", while standardized for many things, can vary through municipalities and depend on specific location, materials available and suggested, etc.
Certainly "Membranes" are usual on either block or frame construction, JUST INSIDE the exterior wall or most often OUTSIDE (On wood frame) to barrier against moisture to a degree. To add "Tyvek" or similar to the outside surface of an interior wall (facing the insulation, is ill effective, and most often Batting/ Paper covered, rolled insulation is barrier material anyway. You'd do well to seal the existing floor, even lay a barrier, and construct a floor above that.
Insulation BOTH WAYS??? Hmmm??? I'll assume you mean roll type? I'd do blown in, in any case, but the issue is what R values ANY insulation will offer, especially if a garage room has no room over it, only roof or crawl space/trusses.
Anything else you do is just as you would for any interior room in the house. You say this is a conversion, and from an existing garage, but you don't state if the existing walls and ceiling of the space are OPEN to the studs, or finished IE: Drywall. You'll also,,, I assume, remove the garage door, re frame that area, and finish appropriately.
45 plus years as a contractor