Royal Damascus was a mark used by Ancien Etablisment Pieper of Herstal, Belgium, c. 1890-1914. T Barker is usually dismissed as "just another Belgian clunker", but with a silver escution and engraving, yours is probably better than the average utility shotgun from the period. I have heard that T Barker was a hardware outlet in California, but have no proof of that.
Simonis, Janssen & Dumoulin, c. 1884-1905, in Belgium
A couple hundred dollars.
10-100 usd
The Benelli Supernova does not come with a damascus barrel as a factory option. However, some custom gunsmiths may offer aftermarket damascus barrel options for the Supernova.
700.00 dollars
50 bucks in good condition
Most likely was imported from Belgium c. 1900-1910. "Nitro" refers to smokeless powder, so it would have been made after 1900. Damascus barrels would place it before WWI.
No published history on the company.
A damascus barrel is a barrel made of damascus steel, which is steel produced by hammering two different steels together. The twist refers to the pieces of steel being twisted while they are hammered, which produces the distinctive pattern on the steel.
You don't.
Not.
Damascus barrel (in French).