How does a nose down stall occur?
Does normal laws apply when the aircraft nose is vertically down?


Yes, always. Physics doesn't care much about the aircraft's attitude and cannot be switched into different modes.

While the nose points straight down, a stall is hard to achieve since no lift is needed - the weight is balanced by drag and inertia. However, when you pull abruptly and start to pitch up, the aircraft can stall while the overall attitude is still partially nose-down. If the pitch inertia is small, the rotation about the lateral axis will increase the angle of attack quickly while the flight path is still more or less straight down. At this attitude the stall angle of attack can be exceeded and the aircraft will stall. This is not unusual - pilots are routinely warned not to pull too hard after ending a spin. And yes, entering the next spin is entirely possible. This is especially likely when the sideslip angle has not been set to zero after ending the first spin. In most cases, this second, unintended, spin is in the opposite direction from the first spin because the pilot maintained his anti-spin rudder deflection for too long.

There have even been cases of supersonic spins.

A few details will help, though. A high pitch rate will delay the stall to an angle of attack up to 50% higher than the stall angle at zero pitch rate. Flying faster will increase the Reynolds number, so stall is slightly delayed when compared to the stall conditions at 1g. Both effects, however, will increase the wing root bending moment, and pulling out of a dive will run a real risk of breaking off the wings when you pull too hard.