The Java Console does not give any error.
What about the other logs files I mentioned? If you don't know what those are then you will need to learn about them if you're serious about supporting OS X.
Please let me know, whether the Apple Java and GNU Classpath will be fine for signed applet made with Sun JDK?
I already said that it should work; definitely on OS X. If you want advice on where to focus your efforts, then I'd say to forget about GNU Classpath (and JVMs such as Sable and Kaffe), and test the Sun JDK for Linux instead. The installed base of GNU Classpath is just way too small to worry about (and it's not even a complete implementation of the class libraries).
I have used Theme with JDesktopPane and JInternalFrame in my Applet, is there any issue with this?
What do you mean by "issue"? If it works as you expect it to, then I'd assume there isn't.