I assume these swirls scratches are in places other than where the stove will sit. The staining looks fine so the swirls are likely caused by the finisher buffing between coats. They may have used a screen that was too aggressive to be used between coats or they may have been using the coarser grit screen to remove dust or something that may have been blown in and embedded in the first coat. Another possibility is that the urethane was put on just a bit thin. A less likely issue is that the floor was tack ragged with something that stayed in the small swirl scratches and reacted with the finish. I've seen swirls like this before and they usually are removed by applying another coat. They should use a finer screen and then maybe follow that up with a Bona maroon pad. I've never seen a finish that was completely perfect, free of dust, hair, a bubble or some other imperfection, but there's a line between what's acceptable and what's not. If the finisher had enough skill to do the nice stain job he should be able to get the finish right.

Flat or matte. I would not use satin or semi. But I dislike sand texture immensely. Impossible, nearly, to wash. I textured over the one room in this house that had sand texture o/ plaster coat. On the ceiling, I could tolerate it but not on the walls.

The secret to both is cream and sugar. If you want a real caramel swirl, use a caramel sauce not caramel candy, which is very solid and sticky, and sometimes brittle when frozen. Add cream, which, brcause of the fat, makes it harden at a lower temperature. There are any number of recipes out there, including specifically for ice cream. Same with the peanut butter. Peanut butters processed differently behave differently, so there may be some trial and error involved, and there are also recipes specifically for swirling in ice cream. Sugar neutralizes some of the sticks to the roof of your mouth-ness. Cream does the sauce for ice cream thing. Food gums might also work, but you have to be good at using them, and like eating them.

Sure you can paint it. My present house and my 1st house has swirled wall and ceiling in every room except the kichen and baths. I painted every room myself. You don't need any special rollers or brushes. You paint as you would a flat wall. If it is unpainted use a primer first. More than likely the swirls are done with dry wall mud. You can use a water base primer for drywall. You don't need oil base. Unless you plan on soaking the ceiling extensively with the primer and scrubbing the primer in the ceiling with a scrub brush the design will stay put. It's harder than you think. The paint store will advise you on the proper product. As with any paint thin coats are better than one heavy coat. My hubby use to do these various textured ceiling for a living in his younger days. The only problem with swirls is if they have to be repaired. It is nearly impossible to match the original swirls. They are fast becoming a lost art as no one has the skill or talent to execute them and they are costly to have done.