Future site of a pending but imminent storm over the definition of "relational". Get your ThreadMess goggles ready...
No need for a storm, or even a breeze. Wikipedia has a fine definition. See http://en.wikipedia.org/wiki/Relational_model
All the ThreadMess this topic needs has already been, uh, messed up at http://en.wikipedia.org/wiki/Talk:Relational_model
Based on that, it does not appear that only highly normalized tables qualify. Normalization techniques were an add-on to Codd's original ideas anyhow.
It is not a matter of "qualification", and the higher levels of normalisation were a result of further research. However, with each level of normalisation it is proven as to what anomalies are eliminated. The higher the level, the fewer the anomalies. Only failure to achieve 1NF would be considered in violation of the RelationalModel per se, and even that is subject to some controversy.
Some interesting info about the relational model:
Indeed, Wikipedia is not always an accurate source, but the summary of the RelationalModel is sound. More comprehensive sources are, of course, AnIntroductionToDatabaseSystems and TheThirdManifesto.And if you come back in a month, it may be all different.
Possible, but unlikely. Technical subjects tend to be fairly well monitored. It's biographical information that tends toward bogosity, as it's considerably more difficult to verify than established technical matters. However, if you're concerned, and are looking for the original -- and arguably definitive and fairly minimal (fewer relational algebra operators are possible) -- definition of the RelationalModel, I refer you to http://portal.acm.org/citation.cfm?id=362685
See also: DoesRelationalRequireTypes