Games are always grouped in the Games hub and don't appear in the app list. Otherwise, there is no official way to group apps except pinning them to Start. I've seen some people use live tiles/pinned web pages as titles for dividing categories on the Start page, that may be a workaround.
There is no way to organize the app list. But you do not need to:
If you have more than (I think) 40 apps installed, the same organization as in the People Hub kicks in; just tap a highlighted letter to easily jump to that part of the alphabet.
There is search button in the top left of the app list which searches for partial matches in all app names. It ...
Games can be pinned to the start screen - simply go to the Xbox Live games app, swipe to the collection tab (if required), and press and hold your favourite game, until the context menu appears -- the first option allows you to "pin to start" - you can also choose to add a review to show in the marketplace or uninstall if you wanted.
AFAIK moving files is not supported as your phone is just recognized as a flash drive, but it actually isn't one - this function is just "emulated". You can "move" files on your SD Card either
by copying the files to the new destination and then deleting the source files. If you run into space problems you could copy the files to your PC first, then delete ...