I have found the various Cisco programs that use Java are very intolerant of other (additional) Java versions being installed on the machine. Especially when Java 7 is installed, I have not been able to get CCP, ASDM or LMS Topology services to run. I also tried selecting Java 1.6 to be the active version for those programs. So I uninstalled Java 7 and install only 1.6.
I am running Flash 11 and that has not given me issues with my CCP. In fact I had to reinstall to correct the "hibernate" bug and the fresh install worked OK with Flash 11. My Java is 1.6 update 31.