iLivid and the searchqu toolbar
Just in case anyone else gets troubled by this:
A member of staff has had endless issues with the Internet, both IE and Firefox on staff laptop (standalone). Browser window appears wrong, searches show up no results, etc. Turns out she has installed iLivid (a download manager) on the laptop, together with the bundled searchqu toolbar. It's the toolbar that does the damage - it resets the home page and seems to interfere with browsing. I couldn't see any evidence of a toolbar on IE or Firefox, but think that she had closed it and that seemed to have made the problem worse as it then became intermittent.
Anyway, you can uninstall it (appears as Windows searchqu toolbar in the Programs list). You'll probably then have to reset the home page on all the browsers too. Helpful info here.
Right - off to have words with a teacher.