Good questions except there is no such thing as a "pinned" barrel on a top-break. (That's a term used with solid frame revolvers.) The serial number should appear on the bottom of the butt, on the rear of the cylinder, between the "ears" at the back of the barrel extension and on the underside of the barrel latch. They all should match.
The ones on the barrel extension and underside of the latch can be hard to see and are quite small. Here's a picture from a 44 DA that shows the locations: (They should be in the same places on the NM#3. The one on the extension may be in two parts. One part on each "ear")