Are there different grades of chicken in the US?
Poultry grading in the US is voluntary, while inspection is mandatory.  Per the FDA:


  Chicken Inspection
  
  All chickens found in retail stores are either inspected by USDA or by
  State systems which have standards equivalent to the Federal
  government. Each chicken and its internal organs are inspected for
  signs of disease. The "Inspected for wholesomeness by the U.S.
  Department of Agriculture" seal ensures that the chicken is free from
  visible signs of disease.
  
  Chicken Grading
  
  Inspection is mandatory, but grading is voluntary. Chickens are graded
  according to the USDA Agricultural Marketing Service's regulations and
  standards for meatiness, appearance, and freedom from defects. Grade A
  chickens have plump, meaty bodies and clean skin, free of bruises,
  broken bones, feathers, cuts, and discoloration. 


The actual US poultry grading standards can be found in the USDA document.  Grade A chicken overall (the actual standards take 5 pages):


Looks nice--no discolorations, marks, bruises, or inappropriate cuts
Has at least 75% of the skin covering it (where appropriate)
Is plump and "well fleshed"
All the feathers have been removed


Any chicken that was actually graded will almost certainly be labelled, as it is a marketing issue.  Otherwise, it is ungraded but inspected chicken.  My personal experience is that most retail chicken, at least where I live, is ungraded.

Factors that affect the toughness or quality of the chicken, in decreasing order of importance, include:


The age of the bird
Any injections or additions to the meat, such as a saline brine, by the packer; brining partially denatures the proteins in the meat, and may contribute to the "rubbery" texture mentioned in the question
The way the bird was treated and fed (some heirloom or boutique chickens are raised in a more traditional manner, where they move about much more, and may have a more varied diet, and  so have tougher, but more flavorful meat as a result--but I don't want to turn this into a political treatise on poultry production methods)
The breed of the bird


The size of commercial chicken is almost entirely related to the age of the chicken; thus the relative toughness of the meat is fairly well correlated with size.