I have both "Region 1" and "Region 2" DVDs that I'd like to view on my PC (Mac). If I have separate external drives for each Region can I watch DVDs of both regions by using the appropriate drive? Or does the OS recognize only one region?
It is a property of the DVD drive, saved in the firmware of the drive. You can usually change it a set number of times (4 or 5, I forget which) before you get stuck with the last region you picked (though apparently you can get the manufacturer to unlock it for you).
I don't know how to change the region with a Mac as I don't have one handy to test with (sorry!) but to change the region in Windows 7 (for people who look at this answer later), follow these instructions: go to the properties of the DVD drive, click the Hardware tab, select your DVD drive there, click Properties button, go to DVD region tab, pick your region & click OK a couple of times.
It is possible to get around the region restriction, either using appropriate software (on Windows, AnyDVD can do it) or by flashing the firmware of the drive to remove it.
I've never had to flash the firmware to remove the region coding, but this blog article gives a nice overview from a Mac perspective.
There are also number of other questions related to removing the region from a DVD drive, most of which are reachable from Disabling the Driver Region on a Mac. Some of them claim that VLC will ignore the region, but I'm not sure as I've never tested it.
To answer your ultimate question, if you have multiple DVD drives then you should be able to assign a separate region to each and pick the right drive for the DVD you want to watch (but personally I would try to get around the region restrictions myself first).