A couple of comments, thoughts, questions:
1. By looking at your cut pictures....the pierce hole is incredibly large. This indicates to me that your pierce delay time is too long. You should be using the pierce delay settings listed in the Hypertherm manual on the "Mechanized Torch" cut charts. Make sure you are using the best quality cut speeds...and make sure you are using the mechanized consumable part numbers....even if you are using a hand torch. The machine shield must be used...not the hand shield.
2. You did not make it clear regarding whether or not you have a height control system.....or if you are just using a fixed height. Excessive bevel is caused by a. Cutting too fast, b. Cutting too high off the plate. If you have a height control....use the suggested arc voltages (listed in the Hyperthem manual) as starting points...then adjust the voltage up or down during the cut to achieve the cut height (.06" for the 45 amp process) listed in the manual.
3. There will always be some taper with an air plasma cutter. If it is consistent around the perimeter it means your consumables are in good shapeand the the height is constant. You can minimize taper to a certain extent by cutting at slower speeds....although too slow will produce low speed dross and warpage. when you experiment with slower speeds...you will need to experiment with arc voltage (if you have a height control) to maintain the correct physical standoff of the torch.
Attached is a picture of a few 1/4" cuts with the Hypertherm 45 amp process.
Jim Colt Hypertherm