I think the hairspring looks ok. In the static state the coils appear concentric and I am of the opinion that the rather weighty stud explains it,s apparent lack of flatness.. I am wondering if there isn't a problem with where the hair spring is attached to the stud. if it is not formed correctly there, I think it would move the h/s off center, causing you problem.
With the movement out of the case I would first check that the stem pieces do actually clip together correctly.
This might be easier by removing the female part of the stem from the movement and holding it in a vice whilst you then apply the male part of the stem with crown. Do they clip together as expected?
Yes? Now put the female part of the stem back in the movement; check it is properly engaged / secured before proceeding.
Now place the movement in the case. Look along the case tube ... is the stem nice and central? Put the case flat on your workbench and try engaging the male part of the stem with crown. You must push evenly along the axis of the stem; do not push at an angle. What happens?