Thank you for the extra pictures. Your gun has been rode hard and put up wet. It was heavily buffed or sanded at some point in time and something was put on the barrel...aluminum paint? There's also some indication that the barrel to frame was worked on or the barrel was removed with a pipe wrench as evidenced by the gouges in the barrel at that junction. Although you might be able to shoot it, I suspect any competent gunsmith will advise against it. I advise against it. The actions on these early guns are long obsolete and finding replacement parts is nearly impossible. I recommend soaking it, without the grips, in a bath of equal parts auto transmission fluid and acetone in a sealable, chemically impervious container for a week or so, then go over it with a brass brush, bronze wool or a copper Chore Boy pad to remove all the remaining rust and grime. Then, spray it down inside and out with aerosol carb/brake/parts cleaner to remove all the oil and grime. I would spray it down, inside and out, with Strike Hold cleaner. This is a special chemical formulation that bonds with the metal and is hydrophobic. That will protect the gun from further rusting. Then I would put it in a glass top wooden gun case or shadow box along with the factory letter and some of the more vintage ammo. Then, the whole family and visitors can admire it and it won't be hidden for decades in a gun safe.
Just one additional recommendation. If you already know this, forgive me. To remove the grips (do NOT pry), completely loosen the grip screw but leave it inserted. Take a wooden or plastic handled tool and tap on the screw head to pop off the right panel, then you can remove the left panel by pushing it off from inside. When you get them off, you will see the grip pins that insert into the bottom of the panels to help hold the grips in position. If you pry, they will break the bottom off the hard rubber grip panel. If tapping the head doesn't work, hold the gun by the barrel and cylinder, then use the tool handle and sharply rap the grip frame at the knuckle to vibrate the panels off. Once you get them off, check the inside of the right panel for the scratched on serial number on the gun. That indicates the grips are original to the gun.
Again that is a terrific inheritance! Thank you for sharing it with us!