Regarding Java, you may have a broken symbolic link etc.
Go to this site and follow the instructions for installing Java .... then you will be able to install java apps like Limewire.
Just pay attention to the version(s) as I think the latest is 1.5.0_10
Also, remove the old symbolic links (javaplugin(s)) before you create a new one.
Do a search for libjavaplugin_oji.so
$ locate libjavaplugin_oji.so
The site was written for FC5, however, it is still relevant for FC6.
Your flash issue(s) may be due to the version that you have installed?
To install the latest Beta version, have a look at this thread.