Wow! Question: Is that an original or a replica? Question 2: If an original does the rear sight have a "C" or an "R" on the side? If the carbine is an original, and if the sight has the "R" on it, DO NOT replace it with a "C" (for carbine rear). If the rear sight is held onto the barrel with blind (no slots) screws, then leave it alone. If with the "R" and has slotted screws, LEAVE IT AS IS.
The carbine appears to have the later 1877 thick, short wrist stock, which replaced the original, thin, long wrist. The latter were too fragile and broke, being replaced in the fields with the newer type.
What about the sight? Well, I have examined a bunch of the "under-50,000 serial number" carbines, and if the stock was replaced, they almost all tended to have the carbine rear sight replaced with a rifle rear sight. Why? Because many company commanders (Capt. Albert Barnitz is documented with the request) asked for issuance of .45-70-405 ammo vice the .45-55-405,
with a corresponding change to the sights! (Note, that such a change doesn't really compensate for the difference in trajectory due to the difference in barrel length between the M1873 rifle and the carbine, but they thought it would!) Sadly, too many collectors have switched the sights back, thus alterring the history of the piece!
OTOH, if you find a M1873 carbine with the 1st issue stock (long, thin wrist), most WILL have a carbine rear sight. This is because the piece probably left government service before the 1879 recall of the >#50,000 S/N for IRAN at the arsenal. Without the later mods, the chances are the piece was sold or distributed to settlers (which WAS allowed by General Order #???...have it, but can't find it right now), OR...was stolen by a deserting soldier!
Regarding your statement about the carbine being a "BLAST!" to shoot...in the words of Astronaut Jose Jimenez, "Oh, I hope not!"
Anyhow, a nice-looking firearm!