Hi,
I may be misunderstanding what your asking.....But try going into the itunes general preferences and set it to use itunes as the internet music playback.
On the problem of not being able to find the song when you quit internet explorer, it may be that you are playing it off the internet but not saving the file. When the quicktime file has fully loaded save as text to a place that is easy to find (a folder on your desktop named downloads ) ect.