Is it correct to say Newtonian mechanics is a subset of Quantum mechanics? I grew up in a three dimensional (3D) reality described quite well by Newtonian mechanics as opposed to the reality described by Quantum mechanics. That is I could go to bed at night without worrying about the bed disappearing during the night plus I could watch the full moon rise in the east before retiring and not have to worry about it disappearing during the night. I was quite confident that it would appear in the west when I arose from my nights rest. Is it correct to say that Newtonian mechanics is a subset of Quantum mechanics?
 A: You are falling into the trap of confusing theories with reality. Theories do not form reality. Reality forms theories. Theories can explain reality, at least partly. It is also quite normal to have different theories co-existing, which explain different parts of reality at the same time. 
The interesting part is usually the borders between them, where unexpected things may happen. But this is simply an expression of the maths breaking down (or being too simple to be applied to more complex cases), it is nothing to worry about, and the fact that different theories describe different parts of nature does not in itself imply that there is a "problem" somewhere.

I grew up in a three dimensional (3D) reality described quite well by Newtonian mechanics 

You certainly did not grow up in the part of reality that is described by Newtonian mechanics, unless you happen to be a celestial body flying through deep space. Newtonian mechanics breaks down at our everyday level the moment you get up from bed. They do not account for 99.999% of the "features" of reality you see around you any second of your live. The parts of reality explained by quantum theories are vastly larger and closer to our everyday lives. 
And there are plenty of other classical theories (electromagnetism, chemistry etc.) which, while not being "complete" to cover the whole Mess[tm] either, are hugely more relevant to us than Newtonian mechanics.
Of course, Newton has the advantage that people can easily understand his rules, and they serve as a great introduction to how to think about problems in a scientific way, while I would doubt the sanity of everyone who claims to fully understand QM... 
A: Newtonian physics is described perfectly by classical mechanics equations.
For dimensions where the Heisenberg uncertainty has a definitive value quantum mechanical formulations have to be used. Macroscopic matter, a ball for example,  is composed by ~10^23 atoms or molecules . Atoms and molecules are in the quantum mechanical regime and bulk matter rests on this underlying framework.

Is it correct to say that Newtonian mechanics is a subset of Quantum mechanics?

The correct  expression is that the classical regime emerges from the quantum mechanical regime . Emerges covers the complicated mathematical processes that are involved in going to the limit of large dimensions and integrating over the many variables which are involved in describing macroscopic matter by its elementary constituents, so as to get the classical mechanics equations.
A: For large-scale physics, Einstein's General Relativity reduces to Newtonian theory of gravitation. Yet for small-scale physics, Quantum Mechanics would lose its indeterministic Nature (the so-called Classical Limit for vanishingly small hbar). 
What could complicate the situation is relativistic velocities which could happen in both regimes. When applied to small scale physics, then Relativistic Quantum Mechanics will be born (which is also called Quantum Field Theory) and it is the most fundamental way to study small-scale Universe. 
Three fundamental forces of Nature, Electromagnetic, Weak and Strong interactions are about to be unified in the context of quantum field theory. But so far, the unification fails to include Gravity. That's why we have two physics; one for small-scale and one for large-scale.
And the reason we are not worrying of things you mentioned in your post, IS NOT that they are not going to happen. It is because that the probability of seeing those weird things in the the course of our lives (which is billions of times smaller than the age of Universe) is vanishingly small. A good measure for those weird things to happen would be the lifetime of proton which IS NOT infinitely long but long enough that we almost will never see it.(The reason proton is thought to be stable is that it has a life time that is at least 100000000000000000000000 times the age of Universe in an underlying supersymmetry model of particle physics.) This is why we think we are safe :) (unless interfered with a Local supernova or gamma-ray burst or even Star Wars!)
Short Answer: It is not correct to say, "Newtonian mechanics is a subset of Quantum mechanics." (Please follow below discussions for details of this conclusion.) 
