Q: I recently moved from MS Windows - often the "Cut" function in Finder is greyed out.  How should I move files between directories?

Try here.

Apple - Support - Mac 101

Switch 101: On Windows, I used to...

Mac OS X keyboard shortcuts

Cut has never been supported in the Finder, because that would require it to be either dangerous or inconsistent with Cut in all other cases.  Instead, if you don't like just dragging and dropping files, try using Copy and then hold down option while pasting (you may note that the Paste menu item changes to Move Item Here when you hold down the option key).

As to #2, I've never seen that behavior.  I can rename files just fine on FAT32 drives. Are you certain that it is FAT32, or might it be NTFS?  The NTFS format can be read by Macs, but not written to without additional third-party software.

Since moving files between volumes requires going back and deleting the file from its source location, yes, two Windows is probably the best way to do it. There's a Finder hack that creates a tabbed Finder called TotalFinder. I've never gotten used to it but lots of peope love it.

As for basic documentation, have a look here:

http://www.apple.com/support/switch101/

If you need to write to NTFS volumes, you might look into a third-party NTFS driver such as that from Tuxera. There is also a free open-source version of the driver called "NTFS-3G" which is pretty much the same thing (without the technical support). Both of them are available here: http://macntfs-3g.blogspot.com

Yes, two windows does make it easy to drag from one location to another.  I tend to like to keep two windows in column view, one in the top half of the screen and one in the bottom.  However, there are other methods as well.  For example, you can grab a file and hover over a folder, and it will pop open after a moment.  You can navigate through your folder heirarchy this way and drop a file in a desired destination.