Feynman makes a point of stating explicitly, in vol. 1 of his Lectures on Physics, that $F = \frac{d(mv)}{dt}$ is not the definition of force. In section 12-1 he states
If we have discovered a fundamental law, which asserts that the force
is equal to the mass times the acceleration, and then define the
force to be the mass times the acceleration, we have found out
nothing.
A bit later he states
The real content of Newton's laws is this: that the force is supposed
to have some independent properties, in addition to the law $F = ma$;
but the specific independent properties that the force has were
not completely described by Newton or by anybody else, and therefore
the physical law $F = ma$ is an incomplete law. It implies that if we
study the mass times the acceleration and call the product the force,
i.e., if we study the characteristics of force as a program of
interest, then we shall find that forces have some simplicity; the law
is a good program for analyzing nature, it is a suggestion that the
forces will be simple.
I found the following comments from Terence Tao, on the topic of how physics models work, to be enlightening:
Terence Tao - @Pietro: the way mathematical or physical models work,
one assumes the existence of a variety of mathematical quantities
(e.g. forces, masses, and accelerations associated to each physical
object) that obey a number of mathematical equations (such as F=ma),
and one also assumes that the result of various physical measurements
can be computed in terms of these quantities. For instance,
two physical objects A_1, A_2 will be in the same location if and only
if their displacements x_1, x_2 are equal.
Initially, the numerical quantities in these models (such as F, m, a)
are unknown. However, because of their relationships to each other and
tophysical observables, one can in many cases derive their values from
physical measurement, followed by mathematical computation. Using
rulers, one can compute displacements; using clocks, one can compute
times; using displacements and times, one can compute velocities and
accelerations; by measuring the amount of acceleration caused by the
application of a standard amount of force, one can compute masses; and
so forth. Note that in many cases one needs to use the equations of
the model (such as F=ma) to derive these mathematical quantities. (The
use of such equations to compute these quantities however does not
necessarily render such equations tautological. If, for instance, one
defines a Newton to be the amount of force required to accelerate one
kilogram by one meter per second squared, it is a non-tautological
fact that the same Newton of force will also accelerate a two-kilogram
mass by only one half of a meter per second squared.)
If one has found a standard procedure to compute one of these
quantities via a physical measurement, then one can, if one wishes,
take this to be the definition of that quantity, but there are
multiple definitions available for any given quantity, and which one
one chooses is a matter of convention. (For instance, the definition
of a metre has changed over time, to make it less susceptible to
artefacts.)
In some cases, it is not possible to measure a parameter in the model
through physical observation, in which case the parameter is called
"unphysical". For instance, in classical mechanics the potential
energy of a system is only determined up to an unspecified constant,
and is thus unphysical; only the difference in potential energies
between two different states of the system is physical. However,
unphysical quantities are still useful mathematical conveniences to
have in a model, as they can assist in deriving conclusions about
other, more physical, parameters in the model. As such, it is not
necessary that every quantity in a model come with a physical
definition in order for the model to have useful physical predictive
power.
Others have gotten into the weeds, so I'll step back a bit. Sorry to break out a mix of maths, philosophy & physics here, but:
it has never been clear what is a definition, what is an axiom, what is a law and what is a proof
Even in pure mathematics, attempting an axiom/definition distinction overlooks the role of axioms as implicit definitions. The axioms of Euclidean geometry, PA and ZFC respectively discuss "points", "natural numbers" and "sets". They don't explicitly define these, but they characterize them by the axioms they satisfy, to the point their names are only of historical relevance, in that these axioms attempt to capture older intuitive notions tied to natural languages' words. Hilbert made a famous comment about this.
A law is presumably a theorem, or stylized variant thereof, satisfying additional criteria I won't try to summarize. It's certainly not a matter of being fundamental, at least not long after a law is named. As for proofs, the real challenge is in choosing what to assume, not in identifying what was assumed or how it has specific consequences. In the empirical sciences, we have an "it works" criterion; the closest parallel in mathematics is consistency, which is much less selective. That's not to say other criteria aren't used for further selection, though.
all my research on these concepts has lead [sic] me slightly in circles.
There are many equivalent formulations of (for example) mechanics that have different domains of easiest usage, which is why they're all worth learning. If you ask "which is fundamental?", none are. One might be the oldest, but "they're all right, because they're all one theory, which is right" is all the justification physics needs. Why are they right? Mathematics can't prove they are, but evidence kinda sorta can (see also Sec. 7.1 here).
I will make what I consider to be the most fundamental definitions... correct me on my "fundamental" definitions... a formal ground-up construction of physics is not something I have ever seen
For the above reasons, this may be the wrong aim.
without given axioms and definitions, nothing can be shown
Not a priori, no. But that's not how science works. Philosophical subtleties aside, it at least occasionally glances at the world to discover its contingent truths.
The goal is to arrive at the principle of stationary action
I realize you have specific goals I've so far overlooked. If you want something deeper that that principle, this will interest you. The basic idea is quantum amplitudes interfere constructively and destructively, and the mean effect is... basically what the classical version of the aforementioned principle says.
they are the result of my attempt to make everything non-circular
One delicious consequence of empirical knowledge is you don't need to worry about whether you're circular . A "concise" characterization of your theory that doesn't repeat itself the way a circular theory might doesn't have any predictive advantages over one that's open to such an accusation. We know we're (probably approximately) correct because the world tells us so.
Axioms based on observation: every body of a physical system has some mass $m$, a resistance to motion, a position in $3D$ space, $s$, and events occur with respect to some order given by time $t$.
It's one thing to say observation warrants such claims; it's another to take them as axioms, or as unique axioms. Ultimately the entire theory is equally corroborated by the evidence, as a whole; data doesn't say which parts to treat as axioms. The role of proofs (putting aside for the moment how deep we have to go to hit "axioms", whose choice may not be unique) is to help us organize the explanation of many observations as the consequences of a few ideas so that (i) if we discover we're wrong (as sometimes happens!) we have a shortlist of what might "have to give and (ii) motivate unifying efforts so we're not just stamp-collecting.
It is important for me to have fundamental definitions of these quantities that are consistent with all of physics, so that I don't get confused when I study those topics later on.
Sadly, we sometimes revise attempts at such axioms when "all" of physics expands. If it works, it works. My main advice for you, however, is to focus on Lagrangian formulations, if only because they've typically had the least trouble adapting in this manner. For example, Lagrangian mechanics accommodates relativity by becoming a field theory, which accommodates quantum effects with operators. Perhaps the biggest upset this will cause your apple cart is the need to focus on canonical, not kinetic, momenta.
the clause "by axiomatic assumption" at the end there is dubious
Why? An axiom can assume whatever it wants. As long as the final theory is neither inconsistent nor at odds with observation, you're fine.
it isn't clear that the definitions of $T$ and $U$ are consistent with each other, in that it isn't clear that they both represent the energy $E$ (which I defined very vaguely...)
If by vaguely you mean implicitly, sure, but that's fine. And if two quantities aren't "clearly" equal, an axiom can say they are, and hopefully that's never false. As I said, I'm leaving other answers to comment on the general validity of your axioms.
it is not really a law (I.e. [sic] a logical consequence of something else) but more just a definition (otherwise what is the meaning of force?)
Oh, you're definitely wrong about that.
Best Answer
The Lagrangian formalism of physics is the way to start here. In this formulation, we define a function that maps all of the possible paths a particle takes to the reals, and call this the Lagrangian. Then, the [classical] path traveled by a particle is the path for which the Lagrangian has zero derivative with respect to small changes in each of the paths.
It turns out, due to a result known as Noether's theorem, that if the Lagrangian remains unchanged due to a symmetry, then the motion of the particles will necessarily have a conserved quantity.
Energy is a conserved quantity associated with a time translation symmetry in the Lagrangian of a system. So, if your Lagrangian is unchanged after substituting $t^{\prime} = t + c$ for $t$, then Noether's theorem tells us that the Lagrangian will have a conserved quantity. This quantity is the energy. If you know something about Lagrangians, you can explicitly calculate it. There are numerous googlable resources on all of these words, with links to how these calculations happen. I will answer further questions in edits.