Does the time for a "cause of action" from a creditor begin when you stop making payments? For example, if I had a credit card from company X and made my last payment on July 2008, does that mean that company X could have obtained a judgment against me then?

Yes and NO in that order.
The "clock" starts on your last acknowledgement, either by payment or in writing.
As for court action, this takes significantly longer and requires a default notices as well as various warnings of impending action followed by a court claim.

According to my most recent credit report from Experian, I was 30 days past due as of "May, July 2008," which I find confusing. Would it be correct to assume that the last payment was made in May and that I became 30 days past due once it reached July? I may have a legitimate defense against a judgment in Arizona. Although a recently signed bill extends the statute of limitations to six years for credit card judgments, if the time of cause for action had reached three years in May or June of 2011, the statute of limitations would have expired under the old Arizona statute; and since laws that extend the length of time for statutes of limitations are not ex post facto, I could have the claim dismissed in court.