When using IE 8 you can easily switch between rendering modes (See third (and last) pic down on http://blogs.msdn.com/ie/archive/2008/03/07/improved-productivity-through-internet-explorer-8-developer-tools.aspx ) to test IE6 and IE7 rendering as well as the IE8 rendering (Windows only).
For Firefox, I think that multiple versions can co-exist on the computer. If that is not the case, then you can run Firefox Portable (http://portableapps.com/apps/internet/firefox_portable) - for those that are ignorant about portable applications: applications that run from a folder. Portable versons of 2.x and 1.5.x are still available. (Windows / Linux - through wine).
I am not too sure about safari on Windows, however I am sure you can install at least one version of safari.
As for operating system I don't think the rendering should be any different between between operating systems (except for maybe IE6 for Mac). However you have been told to test multiple OSs, so try using a Mac (as it is the pickiest about hardware) then emulating Windows and Linux on top.