In Windows 8, can you use a different default browser for Metro/WinRT apps than for normal desktop apps?

Sadly, this cannot be done. In Windows 8, the default browser (one set to open HTML links) is the only browser that can run in metro mode. If you set Google Chrome as the default browser, IE10 will no longer open in metro mode, it will just open in desktop mode. So the default desktop browser automatically becomes the default (and only) metro browser.