Kent, I am new to this, but believe it or not I can answer some of this.
You have to add a line to your gcode to tell the machine to return to zero. You can do this by clicking on the "edit" button located right above the loaded gcode code on the main screen. Just type in what ever coordinates you want the machine to return to and it will do it every time you run this gcode. On reading your question again, it is only the Z that youa re wanting to re-zero.... That tells me that it is quite possible that your bit is slipping down in the collet. This could be due to heat, not tight enough in the first place, or that you are trying to push into the work too far, which can cause the stepper moter to "slip" and not be on a true reading after slipping. It would be my advice to tighten your bit in your tool (make sure the collet AND the bit are clean and dont have any residues on them). Also check to see how deep you are trying to plunge the bit into the work. You may be going too deep (which is why you are hearing the chatter and jerks).
I found how to edit the safe z, but I coudlnt tell you where I found this...sorry. Maybe someone else will chime in and tell you an easy way to find it.
a feed rate that is too high for the tool you have loaded will cause the machine to jerk and chatter. Either you are too deep, or have the wrong tool, or you are going too fast.. decide which it is and change it.
Hope this helps!
Ken