Your questions are all good ones that you should naturally be asking yourself. They are not the kind of questions for which there are really good, crisp answers. Our current understanding of quantum physics is that experts who care about these foundational issues peer at it from a bunch of different angles, poke it with a stick, and try out different mental frameworks. So I'll just try to say a few things about what these views-from-different-angles look like.
Many modern descriptions of quantum mechanics look more like descriptions of quantum information. Examples:
<PERSON>, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062
<PERSON> and <PERSON>, "A derivation of quantum theory from physical requirements," https://arxiv.org/abs/1004.1483
<PERSON> and <PERSON>, "Quantum Theory and Beyond: Is Entanglement Special?," https://arxiv.org/abs/0911.0695
<PERSON>, https://arxiv.org/abs/1303.6917
Seen from this angle, the fundamental building blocks are qubits. Qubits are described by wavefunctions, but unlike the wavefunctions you've probably been seeing in your class, these functions have a domain that is discrete, like {0,1}. Quantum computers can handle problems, such as efficiently factoring large numbers, that classical computers can't do. This is fundamentally because classical information can be converted losslessly to quantum information, but not vice versa. This "lossiness" is the same kind of lossiness you experience when you give up talking about wavefunctions and talk only about probabilities.
You express some discomfort: "I'm familiar with experiments demonstrating the wave/particle duality of fundamental particles (and other small masses), however it seems to me that a mere few paragraphs are insufficient to completely toss out all of our classical models, and accept that reality may be described by wavefunctions, which (for some reason) yield probability distributions when normalized and squared."
Here are a couple of ways to see that quantum physics can't be tamed or confined to a little corner, where it doesn't threaten classical physics.
You mention small masses. Actually there is no mass or distance scale that separates quantum physics from classical physics. The only parameter with units that is relevant is Planck's constant, and you can't manipulate Planck's constant to get units of mass or distance alone. 855
This makes it sort of inevitable that if you accept quantum physics for electrons and photons, you have to accept it for people and planets.
Another way to see that quantum physics spreads inevitably is to think about what it is in the photoelectric effect that prevents the same photon from being absorbed by two different atoms, thereby violating conservation of energy, since energy E goes away, and energy 2E gets created. Because the wavelengths of visible and UV light are so big compared to atoms, a single photon will normally impinge on thousands or millions of atoms. One way of dealing with this that was originally proposed was to give up on conservation of energy as a perfect conservation law, and instead make it only hold at a statistical level. However, experiments by <PERSON> and <PERSON> proved that this was not the case. The only resolution we can find is that there have to be nonclassical correlations between the different atoms -- if atom A receives the photon, then there is an (anti-)correlation that says B didn't absorb it. This is a specific example of a more general fact, which is that quantum theories don't play nicely with classical ones. It's not possible to have, for example, quantum-mechanical photons that interact with classical atoms. Because of this, proving that one system acts according to quantum mechanics is sufficient to prove that all other systems do, if they can interact with the original system.
Continuing with the photoelectric effect, think about that relation E=hf between the energy of a photon and its frequency. That's really all the <PERSON> equation is. (If your book talks about "the" <PERSON> equation as being the thing with $\partial^2\Psi/\partial x^2$ in it, then that's really not quite right. That's just one special case of the <PERSON><PHONE_NUMBER>
Masanes and Mueller, "A derivation of quantum theory from physical requirements," https://arxiv.org/abs/1004.1483
Dakic and Brukner, "Quantum Theory and Beyond: Is Entanglement Special?," https://arxiv.org/abs/<PHONE_NUMBER>
Kapustin, https://arxiv.org/abs/1303.6917
Seen from this angle, the fundamental building blocks are qubits. 800