32-bit programs will always run on 64-bit windows, unless they need a 16-bit component or try to do some goofy thunking or something.
However, one must also realize that a 64-bit program can only access 64-bit libraries- and a 32-bit program can only access 32-bit libraries.
So, a 64-bit program will require 64-bit plugins- and a 32-bit program (such as Firefox, Opera, etc) will only be able to use 32-bit plugins.
the 64-bit version of Java does not install a 32-bit plugin.