On Windows I used to hold the WINDOWS KEY plus the ARROW keys to manipulate window positions. How do I do that on Mountain Lion?

This kind of functionality is not built into the OS. Specific applications may or may not do something like this, though I haven't seen it myself. I believe there are third party bits of software which add this kind of functionality to the OS. You might consider experimenting with the Universal Access preferences (in System Preferences) to enable some form of keyboard manipulation of the mouse which then could be used to move windows about.

charlie