Yes all the siding needs to come off.
Yes the house wrap gets cut and folded over into the RO. Just make an X.
If I was doing it I would use vinyl J moulding not just flat stock, reason being is unless you used at least 5/4 material it's not going to stick out far enough from the wall.
Flashing is then added to the threshold that goes up the sides of the jambs and is folded down to protect the outside wall.
I would install the brick moulding to the door jams first then install the door using 3" long galvinized finish nails. This way you know it's flush to the wall. I add a layer of caulking behind the brick molding. Once the doors squared up and plumb and nailed in place add Z moulding over the top of the door so no water can get in behind the trim.
There needs to be a strip of 1 X vinyl installed tight under the threshold to fully support it.
Now you can install the J molding.
Since there's no nailing fin there no need for window tape on the sides.
Now install the siding. Make sure to leave room for expantion and do not nail it tight to the wall, as cold as it is the vinyl is contracted.
Any stoop or steps need to be at least 4" below the doors threshold.
Looks like your going to have to relocate that condinsate drain.