OK Dave, I'll give it a shot.
The cylinder of the gun you picture is for a 38 DA. So I believe if it is a S&W it is a 38S&W. The patent dates you list do not correspond to any of the old top break numbers. It is close to one though. The 38 Double Action 4th model has the patent numbers 5-11 1880, 1-3 1882, and 4-9 1889.
Is it possible that what you list as 4-8 is really 4-9 and 1888 is 1889?
The gun has been heavily buffed when refinished and sometimes that can make them difficult to read. Also, the serial # range for 38 DA 4th models is 382701 to 539000. Your serial number is missing a digit. If the number on the end of the serial number on the butt was buffed off in refinishing that would explain it. The number was also stamped on the back of the cylinder, on the ears of the barrel latch, and on the
barrel between the ears of the latch. The last two numbers are quite small and hard to read. Another check of the serial number is on the inside of the right grip. At that time it was either penciled or stamped into the grip.
As far as shooting it with modern ammo, if what I said above is correct then your gun was produced around 1896 or 1897 which would have made it a black powder era gun. It was not rated for modern smokeless powder. Many people here on the forum shoot them with modern ammo as modern factory ammo is loaded very light. Be aware though that it could possibly self destruct and even if it doesn't firing it a lot could stretch the frame and loosen the latch.
cflier

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Mark Connot