When I saved a file in adobe after effects CC, it would not allow me to open the file using cs6 on another computer. Is there any way that I can save the file as something that will open in cs6 that will keep my layers and effects editable (the effects that are in both cc and cs6)?
Sure -- install all the various versions of AE you need between the original project file's version, and keep backsaving the project file until you get to CS6.
You need the original CC (version 12.2). Save as CC in CC 2014 or CC 2015, then open it in CC and save as CS6.