Correct JA, the heat affected area will only be 3 mm from the cut at most.
Laser cutting does leave a hardened cut edge and paint doesn’t stick to sharp edges very well as it thins out on the pointy bit.
I imagine that plasma / laser cutters would not buy cold rolled stock as they don’t want any distortion as they cut bits out of large plates. Which means the black coating will be mill scale, just very thin. Phosphate coatings are more grey, but there is a phosphate and oil finish which would give the painting issues mentioned