What is my wall made of and how can I cut it without chipping?
What these layers are, I know one is drywall, oddly enough its the
  inner layer, but I dont know the other 2.


You have plasterboard, followed by plaster "brown coat", followed by finish plaster "skim coat/color coat"


  How can I repair the chipping that took place? Is it just spackle like
  normal drywall?


Traditional spackle would work


  is there anything I can do to keep the chipping from happening in the
  future?


Score the paint film with a knife where you intend to cut, then cut on the inside of that score line. Use a saw with a finer tooth count. The normal coarse-toothed drywall jab saws are for work on unfinished drywall, where mudding, taping, and finish work hides the rough edges. Painted wall surfaces will be prone to flaking and chipping out with a saw like that. Next time use one of these: