It sound like branches are good fit for your problem. Make the presentation. Make new branch in Document->Setting -->Branches. Make the changes for the students, change the document layout to handouts. Make a new branch and add the extra for the teachers. Choose the right branch you want when making the different handouts and the presentation.
Some addition info can be fount in the lyxwiki: http://wiki.lyx.org/LyX/Presentations http://wiki.lyx.org/LyX/HumanitiesLyX#toc4 hth, Ingar