No, C# doesn't have this concept. Each source file is independent in this respect. (And if the using directives are in a namespace declaration, those are independent from other using directives in peer namespace declarations, too. That's a pretty rare case though in my experience.)
You don't need ReSharper to change what gets included in a new class though. You can use the Visual Studio templates.
EDIT: Just to clarify the point about using directives within namespaces, suppose we had (all in one file):
using Foo;
namespace X
{
using Bar;
// Foo and Bar are searched for code in here, but not Baz
}
namespace Y
{
using Baz;
// Foo and Baz are searched for code in here, but not Bar
}
Usually I only have one namespace declaration in a file, and put all the using directives before it.