Does free electrons from the conductor itself flow or electrons inside the battery flow?
some atoms have more electrons than protons?


No, they have the same number but some of the electrons have more freedom to move.


  Does this mean the free electrons from the conductor itself (copper) flow?


Yes. They jiggle about randomly relatively fast (~2 x 10^6 m/s). In the presence of an electric field, they very slowly drift in one direction ( ~1 m/h).


  wouldn't copper run eventually run of its' free electrons?


No, as many are added at one end of a piece of copper as are lost at the other end.



See


How fast does electricity flow?
Do electrons actually flow when a voltage is applied?
Understanding Electrical Engineering with Analogies
How electron movement produces current,instead of having a slow drift speed?