it is very very easy to do what you want. If you get an IDE external USB drive enclosure (which are super cheap these days), you can plug this into any computer and read from the drive. I highly recommend such external drives for backup purposes. Just get one that you can easily open up and put your drive into it (ie dont buy one of those seagate onetouch things with a non-openable built in hard drive). The idea of putting your old drive in a usb enclosure to access it from new computer is a good one.
I've been happy buying such stuff from
www.Directron.com but
www.newegg.com is a great place for all hardware.
As to your second question about running from the external drive without installing on the pc -- whether you can do this may depend on the software you care about, HOWEVER -- keep in mind that the usb drive will look just like a regular drive so that any client who can access the main computer will be able to access the external drive, so i'm not sure your network admin will be satisfied with the solution..