We use some of these terms in the U.S. too, but they are less common and/or mean something slightly different. Some carpenters and millwork companies will call casing architrave instead, but this seems be mostly used for classically inspired, high end work.

I think I've heard some carpenters call baseboard skirting, but the most common usage of "skirting board" is restricted to the baseboard like piece on the sides of stairs.

We have a term cove molding, but it is a particular style of crown molding.

A rail at 3/4 height we also call picture rail. I don't think I've heard of dado rail though.