Were there any traditions that sprang up around a Confederate independence day, or was there even such a day--did the states each celebrate the date of their own secession? What about July 4th--did they celebrate that during the war, or was that too Northern?
I am not entirely sure about the celebrations part of your question....I recall Shelby Foote mentioning that they indeed marked the anniversary of the secession in his "The Civil War" series, but I don't remember how exactly they did so.
However, when it comes to your question about July 4th they would definitely not have ruled out celebrating it, in fact quite the opposite.
You have to understand that, to the Confederates, they were not trying to rid themselves of their American heritage and start anew, but were instead continuing on the legacy of the Founding Fathers. To them, it was a Second American Revolution, and THEY were the rightful heirs of the American tradition, and NOT the Union. This sentiment is illustrated by the fact that the Seal of the CSA features George Washington on horseback.
Hopefully that makes sense. Sorry I could not provide more detail about the celebrations, though.