Do AutoCAD 2012 customized menus work in AutoCAD 2013 software?

yes possible.

while you are using auto cad ver 13

Type on command line "menu"

select the auto cad ver 12 menu 

then auto cad menu will be loaded.

Customized menus can be used. 

Thanks for invitation -

Not sure but I think for any copy of autocad there are some difference - I don't check that before    !!!

I didn't Get Study The Auto Cad and Embedded Software for Programming.      But  about Myself I can Do the work on it  Well.

Yes, i think customize menu of Auto-cad 2012 works in Auto-cad 2013.

yes- customized menus of autocad 2012 should work on autocad 2013

Yes, I do not think there was a major revision to how the .cuix files are defined and used.  There were changes several years ago when .mnu files became .cui and then .cui became .cuix.  I believe the menu file definition is now pretty stable, and you won't have problems.