You shouldn't have any compatibility issues, but technically it's not legal to install the software on more machines than you have liscences for. If you buy the student and teacher edition of office, you are given three liscences :lol:
I know no legal way of obtaining a product key (aside from calling microsoft) for a peice of software that isn't already installed on your computer. If, however, office is already installed on one computer, I'm sure the key is obtainable through the registry.