Is (kg/m) over s^3 a unit and what does it mean in physics?

Answer 1

(kg/m)/s^3 or kg/(m-s^3) is the 3rd time derivative of mass per unit length or the 2nd derivative of mass flow rate per unit length (e.g., liquid forced through a slot). These are rather forced examples. I have no other idea of its physical significance.
Re answer 1: kg*m^2/s^2 = mass*v^2, a unit of energy, not power. Your source is at fault; they give the same base units for energy and power.

Answer 2

Well, since one Watt is a kg*m^2/s^2, your unit is equal to W/m^3, which is the unit of "volumetric heat release rate."