A couple hundred dollars.
Under $200.
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.
Less than $200.
Pieper was a Belgian gun maker, so it was most likely made in Belgium. Look for the Belgian black powder proof marks.
Those early Belgian guns are generally considered wall hangers. Most have a form of twisted-wire barrel construction. If yours is actually 'Fine Damascus' and not actually laminated steel, and if the engraving is better than typical it might bring $200-$300. If it is the standard JABC (just another Belgian clunker) it will top out at $150 as a mantel decoration.
1850-1870
No.
$100 to $150. A little more if it is in excellent like-new condition. A lot less if it is broken or rusty.
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.
Damascus barrel (in French).
Not.
You don't.