You say budget is an issue as well as ease of use. I think you really need to take a hard look at VNC. Especially UltraVNC (»
ultravnc.sourceforge.net/).
You can configure to access your ADS domain for access privs, assuming you use ADS, and options to show/no show icon in the tray. As far as picking from a list of computers, simply create a folder on the teacher's desktop with .vnc files saved in it. Just double click the system name like computer1.vnc.
Really, it couldn't be much easier. Just requires a few minutes setup time for support person.