I would think you would be fine using the .410 adapter. It contains all the pressure.
I would agree, the barrels would be fine, but you might have to keep an eye on the breech face. It should be ok, but "old Damascus barrel shotguns" range from high quality guns to cheap junk, and I don't know what you have...
The pressure of a .410 is within what the gun should handle, BUT its applied to a smaller area than the 12ga so, if the breech is a tad "soft" you might have an issue, eventually.
And many folks shoot older guns with twist and Damascus barrels; some guns are from the 1880s and do just fine with the proper ammo.
This is true. But note that the "proper" ammo is loaded with black powder, and the old style wads. NOT modern ammo loaded to "black powder pressures"
And note that while most of those very old guns do just fine with the proper ammo, every now and then, one of them does blow up. (barrels burst/unravel)
The Damascus method, hammer welding strips of steel together makes excellent knives and swords, but knives and sword blades are not put under the same kind of stress as a gun barrel.
As previously mentioned, Damascus barrels (and twist or Stub Twist) always have the potential of having rust formed inside the voids of the welds, weakening them.
There is NO way to see this, and no way to tell, short of destructive testing (which kind of makes it a moot point)
The older the gun, the longer its been exposed to rust, so the greater the potential risk. There is no way to tell, by eye, SURFACE rust is not the issue, though its a good indicator, but even a gun that looks great can have weakened welds in the barrel.
As to the barrel adaptor, if it could be fired outside of the gun, without letting go, then it won't let go inside the gun, either.