No point in pulling the prop through to check compression. I believe that on some engines that was considered normal, but the reason was to let oil out of the combustion chambers ( Gypsys, Wrights and the like.)
Look to the POH for what to check- which covers all the normal things, or your local flying school/aero club. I reckon anything not in the manual is really the territory of a LAME to be peering into.