Almost all U.S. LTL carriers use NMFC freight classes, although some are attempting to move to a dimensionally-based system.
No. If you claim that your freight is a lower class than it actually is, your carrier will likely find out. They will charge you a fee for reclassifying the freight and charge you the appropriate amount for the actual class of your freight after it has been shipped.
No. While density is one of the primary factors in determining NMFC freight class, other factors, such as value, are also used in making that determination. Gold bars, for example, are very dense, but they fall into shipping class 500 along with very low-density items because of their high value.