Your requests are not unreasonable. In my mind, industry standard is that you should get a full coat of paint (no missed spots and complete coverage with a good film build) and the surfaces should be smooth. You are right about the environment in which the cabinets were sprayed. Some guys go to great lengths to create a dust free area in which to spray and others are cabinet makers first, painters second. With lacquer because it dries so fast, you really have to spray an item and move it out of the area. If overspray hits a door that is already dry, then that door will be rough.

HDF is high density fiberboard as opposed to medium or low density fiberboard. As klem1 wrote, those gaps are excessive and should have been remedied by the manufacturer. Reason number one is any kind of fiber board is susceptible to moisture damage(swelling to start) and those cracks can be the start. HDF and MDF are my last choice for cabinets in moist rooms for that reason alone.

My cabinet are painted with SW proclassic. I keep some in a jar and a small artist brush in the kitchen for touch ups on hard used areas when needed. You can't tell where they were touched up.

I agree with Owl and your painter. It's already been sanded and primed. Inspect everything when they return to be sure you can't see any brilliant blue in the cracks but I think you're good.