How do I integrate an external browser when I want to run and debug my wesite directly out of PHPEd? (e.g. Firefox, Internet Explorer, ... instead of the built in Nusphere-webbrowser)

so-called NuSphere embedded web-browser is actually IE.
To work with system-default explorer (whatsoever is installed as default) you only need to check "Run in External browser". See IDE tab in the settings.