I would start with this ARTICLE, and see exactly what can be done to tune the OS, and the computers for PrE and video editing,which is very, very taxing on all system.
Now, depending on the lab's network setup, there can be major problems with video editing over a network. For those, Adobe T/S will be a good bet.
As Bill says we get very few reports of success using PRE across a network. It also doesn't help that PRE8 was generally considered a bit of a disaster - even when running standalone.
You don't mention the configuration of your hard disks - they can also aggravate problems if you don't have a lot of free defragmented space (minimum for success is 30GB - I go for 60GB).
Another PRE8 issue was the auto-analyzer that takes over most of the CPU activity.
You say that your Tech Support have installed all 'updates' - ask them to confirm that this includes the latest graphics and sound drivers from the manufacturers web sites (nVidia in your case for graphics). And the latest Quicktime player.
And do work through the article Bill referenced.
I'd also be inclined to query with your IT suppliers why, for new HP machines, they provided an out of date version of PRE. PRE10 has been out since 19th September 2011. And, assuming you procured educational licenses, I'm surprised that's not the version you received.
Maybe, to alleviate the frustration of both tutor and students, you could teach some basic principles with something like Windows Movie Maker.
Also download the PRE10 Trial version and see how that works on your systems.
Insanity is hereditary, you get it from your children
I think these articles are about Premiere Pro... but the concepts are the same
PPro and Premiere Elements must run on a single computer, with program and all files on that computer's hard drives
You should have PrEl on the boot drive, with Windows, and all video files on your 2nd 7200rpm hard drive... using only one hard drive is not going to work very well
You should NEVER partition a single drive and attempt to use the partiton for video files
Win Server is NOT supported http://forums.adobe.com/thread/851602
Not in a Network environment http://forums.adobe.com/thread/771151
-and not on a "domain" http://forums.adobe.com/thread/858977