Ah... I think the problem is Prince is starting some GUI frontend that does not exist (as far as I know) in linux. The plugin should be calling a command-line interface, not a GUI. When I click the "Convert" button, I see the "working" progessbar, but nothing else; when the prince process ends, the "Convert to PDF with Prince" window is closed.
According to the
Prince doc, the command-line should work in Windows too, can you configure the plugin and select the prince.exe executable as indicated there?
As for the two taskbar items, I have no idea why that happens, The "Convert to PDF with Prince" dialog is just a plain QDialog widget.
What I don't like from your screenshots is that the plugin icon is missing (from its main window and the calibre toolbar), and the window titlebars are all wrong. Maybe I'm doing something wrong, or maybe that's a common problem with PyQt in Windows...