complexity classes defined as limits or "globally"?
Complexity classes are (usually) expressed using Landau notation, which makes precise what you state. See also our reference questions.

Yes, by definition (usual) complexity classes don't say anything about the real world per se. That is by design; complexity is a model for classifying the inherent hardness problems, not a tool to design or analyse algorithms.
See also here for some related thoughts.


  Is there a way to talk about complexity for finite n?


"Complexity" and "finite" are mutually exclusive. Of course you can study, say, the (relative) performance of algorithms on real(istic) data. The machinery of complexity theory and, sadly, "Big-Oh"-driven algorithm analysis will be mostly useless then.