At present we are at the following level of mathematically modeling interactions of matter:
1) the underlying framework for all matter is quantum mechanical, all other theories will be emergent ( gravity has a question mark as it is only effectively quantized). This framework is evident in the microworld of particle physics and is important as long as boundary conditions are with small distances and $h$ ( Planck's constant) in the uncertainty relationa $[x,p] >h/2π $ is fulfilled non trivially, i.e. the measurement accuracies are small enough . If $h/2π$ can be considered zero within measurement accuracies, one is in the classical frame.
2) The classical framework theories are emergent from the underlying quantum mechanical, and in the region of overlap it can be shown mathematically that the known classical equations hold non trivially. This means there is mathematically a smooth transition to Newton's mechanics
Newtonian gravity laws hold until masses and energies reach very high values.
c) For high velocities special relativity describes the data, and for high energies and masses the framework of General Relativity fits all cosmological observations to date.Again in the overlap region General Relativity reduces to Newton's law. Please note that both General and special relativity are used in the GPS system that is so wide spread over the world .
Is it fair to apply Newtons law using the weight of a student and the weight of a grain of dust (and explaining that in reality other forces will totally cancel out any measurable effect) etc?
Yes, because if you put in the numbers, neither quantum mechanics nor special and general relativity are needed, the classical formulae are well within the errors.
And does the law really break down on the extreme heavy objects as claimed in the video?
Break down if used in the sense that the Newtonian formulae do not predict the correct numbers , yes. The situation has to extend to another framework
For particle physics special relativity is necessary, but models still use the gravitational potential even if it has little meaning for such small masses. It will depend on the particular problem and the boundary conditions . Since quantum mechanics is necessary for particle interaction models, it is fair to say that Newtonian gravity is not easily modeled.
Theortical physicists are working towards a theory of everything where gravity will also be definitevely quantized, the the transition between frameworks will be mathematically predictable. At the moment string theories can have quantization of gravity and embed the standard model of particle physics, but the definite theory is not yet attained. In any case for everyday classical experiments there will be no difference when it is attained (imo of course).