The style of forged handrail with rings across the top (circles in a horizontal line just below the top rail and above the balusters is fairly common. I'm trying to figure out if such a design would have been used in the 1700s and/or 1800s. Does anyone know if this style has a name? Does anyone know whether this style was used 200-250 years ago, like late 1700s/early 1800s? 

That pattern is usually just called a “ring band” (or ring panel) railing

The Historic Blacksmith Conservancy recently posted a video about a set of French books from the 1920s on Ancient Ironwork.  As he was turning the pages of the Louis XV and Louis XVI period and some of the railings appear to have this pattern, but it's difficult to tell in a frame or 2 of video.