Ok, I'm fairly new to utorrent too but a few questions to try and help us.
So the files don't remain in the window and say "seeding" in the status column?
Try the very last paragraph in this post first - that may be the answer. If not, read on
Do they remain in the same folder you downloaded them to or have you selected for them to be moved to another folder on completion?
I don't know if uTorrent will seed if this happens (Vuze won't if the file is moved), I haven't tried it.
I don't see any box that tells it not to seed when finished so I'm as stuck for any further suggestions at the moment.
http://filesharefreak.com/2008/11/02/utorrent-move-all-your-downloaded-files-keep-seeding/
has some ways of reseeding moved files but there may be something in there that may help (like the file utorrent works from which may have been accidentally altered etc).
Oh, one other thing.
In the top left of the uTorrent screen, you can select what you see in the main window (downloading, completed, active, inactive etc).
If you have selected the "downloading" screen, the download WILL vanish when it's finished.
Skip through the other screens and see if they are there and seeding.
If those options are not there, go to options at the very top and make sure "show categories" is ticked - or press f7.
