Fundamental equations I was reading Our Mathematical Universe by Tegmark and thinking about the relationships of physics equations to the five he lists on page 250: Maxwell’s electromagnetism equation, Einstein’s special and general relativity equations; Schrödinger’s wave equation, and Friedmann’s expanding universe equation. 
I have two questions. First, can the numerous equations in our physics books (the advanced placement physics equations: classical mechanics; fluid and thermal physics; waves and optics; modern physics; and electricity and magnetism) be calculated from these “fundamental equations? Second question, are there other fundamental equations beyond the five?
 A: You have read that the basic physics equations would fit on a t-shirt. This is true in a very broad sense, but you need to be very careful here:
First of all, let us distinguish a mathematically phrased theory about physics from the world around us. Whether or not the universe "has" a mathematical structure is an ontological question which physics can't really answer - this is metaphysics. What is clear is that we can - to increasing precision - an increasing amount of phenomena in purely mathematical terms. 
Mathematics is a structural science that deals with abstract objects, so in order to actually do physics with mathematics, we always have to put down a number of postulates/axioms to "identify" a physical object with some mathematical object in our theory. 
An equation doesn't do that. An equation only relates objects in a given set of axioms, but it doesn't really tell you what they are. So in order to mathematically describe physics, you'll have to first write down all the postulates, namely set up what you think is the mathematical structure for space, time, etc. Otherwise, any equation is senseless. Using Maxwell's equation, I can derive some wave-equation as it also relates to sound waves, yet I don't think you'd say that the equations for electrodynamics describe sound propagation - because you implicitly identify the objects in the equations by electric and magnetic fields and the wave equation then describes electromagnetic waves.
Usually, what we do is that we make heavy approximations: A ball might become a pointlike object in Newtonian mechanics, we'll always neglect small effects (nuclear magnetic fields are neglected and/or described in "effective" ways when we study the effects of magnetic fields on macroscopic objects, etc.). 
But this is not all: You'll also have to state further postulates/axioms about how the objects you identified behave - how the behaviour is described or how you can obtain the equations of how your objects behave from other principles. For example, you can describe Newtonian physics with an action principle, where the particles will always minimize some "action" of the system that is described in your postulates. Then you will also have to say how to obtain the equations of motions (your "fundamental" equations) from these axioms.
Finally, after making all identifications, you'll have a physical theory. Now you can of course define an "ordering" of these theories, where a theory is contained in another theory, if all it's axioms can be derived from the other theory mathematically. Then you can of course ask the question whether there is one theory that contains all other theories and call it "Theory of Everything". Maybe such a theory does not exist, maybe it does, but at the moment, we have two different theories, the standard model and general relativity, that are not contained in one another. The dynamics of general relativity are described by Einstein's field equations and in principle, the dynamics of the standard model are completely determined by the Lagrangian of the standard model (here is one version of it). However, you should note that this is a very incomplete picture without also writing down all the identifcations (which become very difficult: We cannot really directly see a quark, so it'll not be enough to tell me what type of mathematical object is given by a quark - no, in principle you'll also have to tell me how to build an experiment to actually see this quark so that I can actually compare the theory and reality). 
What is this TOE useful for? Probably not for very much (see CuriousMind's excellent and succint comment). In principle, Maxwell's equations are a consequence of the Lagrangian you see above, but writing down how this works in pracice is far from trivial (nor is it actually fully established, because pinning down all the different relations between the "real world" and the mathematical objects is actually very difficult; in a sense it's just something that all physicists believe in for extremely good reasons). 
I would like to have these mathematical derivations (if a mathematically phrased physical theory of everything exists), because it would mean that our equations for nature are very "consistent" and the mathematician inside me would be extremely satisfied, if this were possible, but we don't learn anything from that in physics (only philosophy).

To sum up: If you are just interested in equations, there might be such a set of equations and you can probably print the fundamental ones on a t-shirt (so far, Einstein's equations and the standard model Lagrangian are as far as you can probably get). But if you actually want to write down the physical theory of everything, making the identifications and writing down the posutlates will probably fill a book. Writing down the derivations of simpler "effective" theories is even more challenging and will probably fill books upon books and in the end, you'll not have learned anything very useful about the world. 
A: 
First, can the numerous equations in our physics books (the advanced placement physics equations: classical mechanics; fluid and thermal physics; waves and optics; modern physics; and electricity and magnetism) be calculated from these “fundamental equations?

No. They're derived from postulates and observation and experiment, not from five magic beans. 

Second question, are there other fundamental equations beyond the five?

No. There are no "fundamental equations". Energy is fundamental. So is space and motion. They are what they are, and we describe what they are and what they do with equations. But the notion that equations are themselves fundamental is just popscience tosh for suckers and kids. As is the notion that the universe is made from maths. I'm afraid that's the sort of outrageous garbage that smirking wannabee celebrities come up with to try to garner media attention and make their name and pimp their book. See this for what is hopefully an interesting read. 
