Windows – Can I run a 32bit version of Windows XP in a Virtual Machine, on a 64bit machine running Ubuntu (64bit)

You don't need JDK to run Vuze, JRE should suffice. Apparently it does matter which Java is installed on the computer. I had a similar issue with 64bit Internet Explorer. So:

1. Yes, it does
2. Yes, it is possible to install both JREs
3. You don't need to pay attention, it should find the appropriate version automatically.

Java is designed to do this kind of thing transparently. Ever heard the slogan Compile once, run everywhere? Well, it's relevant here: Compile once, debug everywhere.

The problem you are seeing is because the GUI is windows specific. The exception you see confirms this. In general, anything that uses SWT or AWT will not work correctly on a different platform unless care is taken to make sure that it does.

Long story short, you're out of luck. You could of course use JD to extract the source and modify the GUI code to run properly on OSX. Unfortunately, that would require some experience with Java.