Howdy
First off, the cylinder on a 44-40 44 Double Action was 1 9/16" long. This is 1/8" longer than the standard 1 7/16" cylinder that most 44 Russian Double Actions were equipped with. The cylinder had to be 1 9/16" long to chamber either 44-40 or 38-40. Compare your photos to these two 44 Russian 44 Double Actions. Compare the proportions of the cylinders to the frames. Secondly according to SCSW, the SN range for the 44-40 version was 1-15340.
Peering into the chamber of a 44 Russian there will be a visible step in the chamber where the chamber narrows down to the chamber throat. But both sections of the chamber will be cylindrical, there will be no taper. In addition to the step for the chamber throat, a 44-40 chamber, or a 38-40 chamber will have a noticeable taper for the tapered cartridge case.
The gun in the photos you provided has definitely been refinished. The hammer and trigger are blued, S&W never blued hammers or triggers, they were always case hardened.
This 44 Double Action has been refinished. Notice the blued hammer and trigger. The old dime is a replacement that I had put on for the front sight.
On this side, the edge of the side plate was rounded over by over polishing. The joint should be almost invisible.
Notice how the outline of the side plate is almost invisible on this nickel plated target model.
All told, for what it is, $600 is probably not too bad a price for that gun. Of course, check to see that it functions properly. Also check to see how good the lockup is at the barrel latch. Many of these guns shot loose over time and there will be a bit of play at the latch when the barrel is latched. Ideally there should be zero play. Check the chambers and bores for pitting. It is rare to find a gun of this age that does not have some pitting in the chambers and/or bore. But I have found that as long as the rifling is still strong, it will still put a good spin on the bullet and the gun will be a good shooter.
I notice what looks to be a repaired crack on the right grip. The old hard rubber grips got brittle over time. Be very careful removing the grips, if you pry at them they can break. And if that repair is not sound it might break there again.
Lastly, and there are those who will disagree with me, I only shoot these old Top Breaks with ammunition loaded with Black Powder.