From: Colorado Springs
AFAIK The Matrix serial nrs only are checked on install - or rather you must have one to install-, but no other limitations, so you can put it on an external HD and play it from any computer, just the reg keys are to change as stated by Larry, if the HD designation (E:, F:,..whatever) changes : edit your TOAW3.reg file and "load it" into registry (dble-click it for that).
Maybe Erik could confirm, but I think all this is perfectly doable and legal.
The serial numbers are actually checked as part of the startup, so there is a chance that if you're playing with a pirated copy, the next patch may not let you play. Matrix scrapes the pirate sites looking for illegal serial numbers. They're also checked to see if both PBEM players have the same serial number, and it turns off the anti-cheating features. I call that one poetic justice.
I'm not sure that you need to edit the registry, as long as it was installed somewhere so that the serial number key is there, I routinely run it on a USB key. Editing and loading may be a bit slower, and you may want to turn off the 'Movies', but I've tried to keep all Disk I/O limited to the startup and the scenario loads.
TOAW III Programmer
My comments are my own, and do not represent the views of any other person or entity. Nothing that I say should be construed in any way as a promise of anything.