There's infinity LED cubes on the web; I don't think I'd be able to improve on them so I won't try, but:
Will it be possible to get the code permanently written on a Micro controller chip so that I can properly case the cube n make it permanent. (I ask this question because I know you can make a copy of chip using the UNO but I have a Mega ADK having a Mega2560 and would like to know if it is possible to make a copy of the project on a smaller chip ie: ATMega 128, 328 etc. )
As long as you don't use more IO pins, Flash, or RAM than the 128/328 has, then yes you can transfer it. Once your project is done you can get a blank chip and burn the bootloader and/or sketch to it, basically you're looking to do this:http://arduino.cc/en/Main/Standalone
Note that you may not need all of those parts -- for example if you're using an external regulated power supply you won't need the 7805.