These are ideas in theory, without any personal practice behind them.
For the stem wall inside layer, you could use a center pole with a radius scraper guide maybe 1’ by 2’ . I’ve seen that used on hand-build stone and mortar silos, to keep vertical and cylindrical; you swing the radius part in a circle to show where to lay the stones, then mortar them using the scraper to make a smooth inner face. Lift the radial spinning part to the next layer, repeat. For the dome part, you would use a ball joint at the top of the pole and a spherical section mortar guide - scraper - smoother, to remove any parts of the inside wall that protrude into the ideal half-sphere.
Using these takes more prep and build infrastructure, but if you are doing multiples, it might pay. A rigid guide like this might also show any distortions in the airform.
I guess you strip the airform then, and add the grid of basalt rope, and spray the outside again. To control the depth of the outside layer, you could to use ‘shooting pins’, taped or glued to the outer concrete layer. A ‘shooting pin’ is a sacrificial (left in place) bit of wire (or preformed plastic?) with a perpendicular part the desired depth, connected to a plane that can be attached to the outer gunnite layer, or to the airform for the inner layer. Consider a paperclip flat on the table, open the first outer bend from 180 to 90 so that the plane part is more stable, twist the second bend out of plane to make the vertical, completely straighten the third bend to make the vertical longer.
I’m not imagining the J hooks and slots description for the door; but here is another idea.
In the US, one of the ways you can buy a door, is prehung with a split frame. The prehung door with the hinge side is attached to rough framing on one side of the doorway, then the split frame slides onto it from the other room side, the split joint is concealed under the 3/8" door stop.
It seems like you could do something similar for your method of ‘spray inside’, ‘spray & cover outside’; Install the interior door frame (or door frame buck at least) on the airform inside, and spray around it. This is also a depth gauge for the stem wall near the door. Remove the airform, attach the outer half of the split door frame, then build around that. I hope that is clear enough.