Is America class structured?

More or less, yes. Though not as a function of any national policy. 

The rich run everything, own most everything, can influence govt directly for their own ends, and have access to better educations, key social contacts, more earning opportunities and a different set of legal standards. 

No.

A class structure would mean you can't move from one class to another.

America is an oligarchy, but it's possible to move between classes.