Thank you all very much for your continued interest.
In a nutshell, what happens with these computers is that they are wiped completely clean and start afresh every few hours from an "image" of the operating system. Thus, there is no history at all, of anything at all, at that point. It is a very quick way for public installations to remove all the junk people like to put on the computers. Here is a tutorial on how to do it for XP:
http://blog.hishamrana.com/2006/02/22/how-to-image-windows-xp-with-ghost-and-sys prep/
Thus, it is important that whatever settings are in the image are actually the ones that the system is going to use. What appears to be happening with QuickTime is that the first time you try to access an mp3 link (which it for some reason associates with an MPEG audio file), something undoes your wish that it NOT be QuickTime, and checks "MPEG audio" for QuickTime in the QuickTime MIME settings. Actually, this morning, it actually took TWO clicks and not just one, before the invalid check mark appeared -- but QuickTime still came up on the second click. Then I went into the QuickTime control panel -- again -- and unclicked the MPEG audio MIME setting -- again -- and this time it took. It would continue to be OK, now, I think, from this point on.
I think why you can do it on your computer, Jacumba, is that you are not starting from scratch -- and that will be hard to duplicate. You are essentially where I am now, beyond the funny business at startup, and everything will appear to work as it should with your selections in the check boxes.
It is only at startup that there is a problem. However, the way these computers are being used, in starting from scratch every few hours, that's a critical step for them. If the system administrator can't make a selection that holds right from the get-go from the image, he is right in saying that "the only way to do it is to uninstall QuickTime and not have it on the system." He spent a lot of time on this and is an extremely competent fellow, so I do not take what he says lightly. He said he does not want to spend any more time on it, and that is why I am trying to see if there might be something simple. I'm beginning to believe he is right.
Unless it is possible to prevent QuickTime from undoing your selections -- right from the beginning, and not at some later point -- the only way students will see Windows Media Player as default, as desired, when they turn on the computers is to not have QuickTime on them.