First, in the Civil War era, the term "Navy" was applied to .36 caliber revolvers and "Army" to .44 caliber, even though an individual gun may have been sold on the commercial market. That is the Starr Double Action Army Model. Some were sold commercially prior to the Civil War, but the bulk of the .44 Army production went to the Union forces.
The grip markings (stamped in, not inked) are the Army inspector's markings, GKC for George K. Charter, a Springfield Armory sub-inspector assigned to the Starr factory, so the gun was in fact part of a military contract and is definitely a Civil War relic. Reportedly, the guns were not very well liked, in spite of what may now seem advantages.
The various letters are factory inspectors' marks.
They are fairly common, some 23,000 of that model being made, and generally bring around $1200 (retail) in that condition, though one like new would probably run over $3000.
The serial number is on the front of the top frame below the rammer. There are no "matching numbers" as on Colts.
They are definitely collectors' items, though yours is not of museum quality unless there is some connection with a prominent person.
I would see no obvious reason that Starr cannot be fired if you choose to do so. Loading instructions and charge would be the same as for a .44 Colt.
FWIW, repros have been made; they are nowhere near the quality of the originals, but they are made of modern steel, where the originals are wrought iron.