Somebody more knowledgeable might chime in, but the .38 S&W ammo looks to me like from the '60's or '70's. It is made for older revolvers; it's less powerful than .38 Special, with a case shorter and a few thousanths of an inch wider than that cartridge. It should not, and normally will not, chamber in a .38 Special cylinder.
It will fit many of the old top-break revolvers, and the .38 K-frames made for the Brits in WWII days. It would be wise to have any old gun checked out before attempting to fire it.
It's loaded with smokeless powder, but to relatively low pressure so it SHOULDN'T damage the old guns. IF IT HAS HAD GOOD STORAGE, it should be safe to fire, if you get access to one of those revolvers and care to shoot it.
I probably would, but you should be extremely cautious about checking for a clear barrel before firing a second shot if you get a suspected squib (less noise and recoil than normal) discharge.
About the .44 Special, it can be fired in any .44 Special or .44 Magnum firearm in good condition. Same caveat about a squib load, more likely if ammo looks corroded, or if it has been exposed to oil or excessive heat in storage.
Again, this info is subject to revision from anybody more knowledgeable.