Can I file direct to MINUS or PICKN and then pick up the rest of the HEEHAW FIVE into BNA? In other words can I pick up a STAR at an intermediate location on the STAR?
You can file 1) MINUS/PICKN.BNA but not 2) MINUS/PICKN.HEHAW5.BNA
For 1) the center computer will probably issue a PAR (Preferred Arrival Route) of the HEHAW5 arrival and appropriate transition, depending upon which direction you're coming from. You can expect an ATCer to see this and issue you the PAR.
For 2) the center computer will not recognize MINUS/PICKN as an appropriate transition to the HEHAW5 so it's doubtful your flight plan would be accepted in the first place.
The only intermediate location this approach appears to accept is TINGS. You can tell this information by the accepted transitions listed at the bottom of the plate.
If so would I just do the Central City Transition? Will that work? Or do I have to start at the Central City VOR or Farmington or...? Or will the ATC guys be like "The hell?"
You should file for any one of the 6 appropriate transitions as listed on the plate. The only thing an ATCer is going to do is issue you the PAR unless your remarks section says "No STAR".