While photos of the gun would be helpful, most of the information in your description points to the weapon being a Colt Special Model 1861. Colt manufactured 75,000 of these for Federal forces during the war, and it was .58 caliber. One of his contracts for this model was secured in 1863, which would account for the date on your piece.
This was a rifled musket similar to the regular U.S. Model 1861 Springfield, except that the barrel bands were held in place by screws rather than band springs, it had no clean out screw at the bolster below the nipple, and the hammer and ramrod differed somewhat. One is now for sale at the Sharpsburg Arsenal for $3,200 with photos at this website:
Most buck and ball ammunition used in the Civil War was a .69 caliber round ball along with several smaller buckshot, and was designed for smoothbore muskets, like the Model 1842 or flintlocks converted to percussion, which had much less effective range than a rifle musket. Some Union and Confederate regiments still carried smoothbores at Gettysburg, but rifle muskets were more prevalent at this stage of the war.
It seems likely that your weapon is a Colt Special M 1861. How did you determine the caliber of the barrel to be .60 caliber? Allowing for an error in measurement, .60 is close to .58. Have you examined the inside of the barrel with a bore light for rifling, or are you just assuming it is smoothbore from the buck and ball ammunition saved by your ancestor? If as I suspect the piece is a Colt Special, neither the powder flask nor the B&B ammunition were likely used with the weapon during the war … but may have been stored with it by your ancestor for some reason. A Colt Special would have used pre-measured cartridges during the war.
The caps would have fulminate of mercury inside them and are not particularly dangerous, even if still functional.
You could learn more about your ancestors regiment (and possibly its small arms) if you could get hold of a copy of the book "157th New York Volunteer (Infantry) Regiment : 1862-1865, Madison and Cortland counties, New York" by Isabel Bracy. While out of print, maybe one is available via interlibrary loan or a specialty bookstore.
Finally, several sources indicate the 157th NY Infantry was armed with Enfield rifle muskets at the time of Gettysburg. At times however, separate companies within the same regiment were armed with different weapons, so some of the unit's companies by midwar may have been armed with Enfields (.577 or .58), some with Colt Specials (.58), and conceivably even some still with smoothbores (.69). A sister regiment of the 157th NY in the same brigade, the 61st Ohio, had "Enfield .577, .58 and .69 rifled" ammunition at Gettysburg. (This may have been the source of your ancestor's B&B ammunition?) Regiments typically improved their weaponry during the war through either later govt. issues or battlefield retrieval.