The problem is not the version, since you're using COM to get AutoCAD. The GetObject calls worked back in VB6 days, so it's not that the AutoCAD.Application.15 object can't be created, it's that you're trying to cast it into a defined type it won't match. IN your project, what assembly do you have loaded that defines the AcadApplication type? I can guarantee you the 2002 and 2013 versions don't play nice together.
I would build this project in .NET 4.0 and have _acadApp be a Dynamic variable.
Dynamic _acadApp;
try
{
Type acType = Type.GetTypeFromProgID(_progID_2002);
_acadApp = Activator.CreateInstance(acType, true);
}
It will mean you won't have any intellisense while writing, and it will compile even if you've typed in methods/properties that don't exists. That means you may have methods that work for 2002 and not 2013 and vice versa. If you have to use .NET 3.5 or lower, Reflection would be the way to go.