The relation is very deep and has a rich mathematical structure, so (unfortunately) most stuff will be written in a more formal, mathematical way. I can't say anything about Donaldson theory or Floer homology, but I'll mention some resources for Chern-Simons theory and its relation to the Jones Polynomial.
There is first of all the original article by Witten - Quantum field theory and the Jones polynomial. A related article is this one (paywall) by Elitzur, Moore, Schwimmer and Seiberg.
A very nice book is from Kauffman called Knots and Physics. Also the book by Baez and Munaiin has two introductory chapters on Chern-Simons theory and its relation to link invariants.
There are also some physical applications of Chern-Simons Theory. For instance, it appears as an effective (longe wavelength) theory of the fractional quantum Hall effect. Link invariants, such as the Jones polynomial, can be related to a generalized form of exchange statistics. See this review article: abs/0707.1889. See also this book by Lerda for more on this idea of generalized statistics.
One of the reasons relativistic theories are so restrictive is because of the rigidity of the the symmetry group. Indeed, the (homogeneous part) of the same is simple, as opposed to that of non-relativistic systems, which is not.
The isometry group of Minkowski spacetime is
\begin{equation}
\mathrm{Poincar\acute{e}}=\mathrm{ISO}(\mathbb R^{1,d-1})=\mathrm O(1,d-1)\ltimes\mathbb R^d
\end{equation}
whose homogeneous part is $\mathrm O(1,d-1)$, the so-called Lorentz Group1. This group is simple.
On the other hand, the isometry group of Galilean space+time is2
\begin{equation}
\text{Bargmann}=\mathrm{ISO}(\mathbb R^1\times\mathbb R^{d-1})\times\mathrm U(1)=(\mathrm O(d-1)\ltimes\mathbb R^{d-1})\ltimes(\mathrm U(1)\times\mathbb R^1\times\mathbb R^{d-1})
\end{equation}
whose homogeneous part is $\mathrm O(d-1)\ltimes\mathbb R^{d-1}$, the so-called (homogeneous) Galilei Group. This group is not semi-simple (it contains a non-trivial normal subgroup, that of boosts).
There is in fact a classification of all physically admissible kinematical symmetry groups (due to Lévy-Leblond), which pretty much singles out Poincaré as the only group with the above properties. There is a single family of such groups, which contains two parameters: the AdS radius $\ell$ and the speed of light $c$ (and all the rotation invariant İnönü-Wigner contractions thereof). As long as $\ell$ is finite, the group is simple. If you take $\ell\to\infty$ you get Poincaré which has a non-trivial normal subgroup, the group of translations (and if you quotient out this group, you get a simple group, Lorentz). If you also take $c\to\infty$ you get Bargmann (or Galilei), which also has a non-trivial normal subgroup (and if you quotient out this group, you do not get a simple group; rather, you get Galilei, which has a non-trivial normal subgroup, that of boosts).
Another reason is that the postulate of causality is trivial in non-relativistic systems (because there is an absolute notion of time), but it imposes strong restrictions on relativistic systems (because there is no absolute notion of time). This postulate is translated into the quantum theory through the axiom of locality,
$$
[\phi(x),\phi(y)]=0\quad\forall x,y\quad \text{s.t.}\quad (x-y)^2<0
$$
where $[\cdot,\cdot]$ denotes a supercommutator. In other words, any two operators whose support are casually disconnected must (super)commute. In non-relativistic systems this axiom is vacuous because all spacetime intervals are timelike, $(x-y)^2>0$, that is, all spacetime points are casually connected. In relativistic systems, this axiom is very strong.
These two remarks can be applied to the theorems you quote:
Reeh-Schlieder depends on the locality axiom, so it is no surprise it no longer applies to non-relativistic systems.
Coleman-Mandula (see here for a proof). The rotation group is compact and therefore it admits finite-dimensional unitary representations. On the other hand, the Lorentz group is non-compact and therefore the only finite-dimensional unitary representation is the trivial one. Note that this is used in the step 4 in the proof above; it is here where the proof breaks down.
Haag also applies to non-relativistic systems, so it is not a good example of OP's point. See this PSE post for more details.
Weinberg-Witten. To begin with, this theorem is about massless particles, so it is not clear what such particles even mean in non-relativistic systems. From the point of view of irreducible representations they may be meaningful, at least in principle. But they need not correspond to helicity representations (precisely because the little group of the reference momentum is not simple). Therefore, the theorem breaks down (as it depends crucially on helicity representations).
Spin-statistics. As in Reeh-Schlieder, in non-relativistic systems the locality axiom is vacuous, so it implies no restriction on operators.
CPT. Idem.
Coleman-Gross. I'm not familiar with this result so I cannot comment. I don't even know whether it is violated in non-relativistic systems.
1: More generally, the indefinite orthogonal (or pseudo-orthogonal) group $\mathrm O(p,q)$ is defined as the set of $(p+q)$-dimensional matrices, with real coefficients, that leave invariant the metric with signature $(p,q)$:
$$
\mathrm O(p,q):=\{M\in \mathrm{M}_{p+q}(\mathbb R)\ \mid\ M\eta M^T\equiv \eta\},\qquad \eta:=\mathrm{diag}(\overbrace{-1,\dots,-1}^p,\overbrace{+1,\dots,+1}^q)
$$
The special indefinite orthogonal group $\mathrm{SO}(p,q)$ is the subset of $\mathrm O(p,q)$ with unit determinant. If $pq\neq0$, the group $\mathrm{SO}(p,q)$ has two disconnected components. In this answer, "Lorentz group" may refer to the orthogonal group with signature $(1,d-1)$; to its $\det(M)\equiv+1$ component; or to its orthochronus subgroup $M^0{}_0\ge+1$. Only the latter is simply-connected. The topology of the group is mostly irrelevant for this answer, so we shall make no distinction between the three different possible notions of "Lorentz group".
2: One can prove that the inhomogeneous Galilei algebra, and unlike the Poincaré algebra, has a non-trivial second co-homology group. In other words, it admits a non-trivial central extension. The Bargmann group is defined precisely as the centrally extended inhomogeneous Galilei group. Strictly speaking, all we know is that the central extension has the algebra $\mathbb R$; at the group level, it could lead to a factor of $\mathrm U(1)$ as above, or to a factor of $\mathbb R$. In quantum mechanics the first option is more natural, because we may identify this phase with the $\mathrm U(1)$ symmetry of the Schrödinger equation (which has a larger symmetry group, the so-called Schrödinger group). Again, the details of the topology of the group are mostly irrelevant for this answer.
Best Answer
The no-go results from Algebraic and Constructive QFT you mention deal with related but slightly different matters.
(Edit: the previous version of the following paragraph was slightly misleading - Haag's theorem is actually stronger than I stated before; see below for details)
This has consequences for both scattering theory and attempts to rigorously construct field theoretical models starting from free fields. In the first case, Haag's theorem is circumvented by either the LSZ of Haag-Ruelle scattering formalisms, which obtain the S-matrix by respectively taking infinite time limits in the weak (matrix elements) and strong (Hilbert space vectors) sense. Recall that both setups require the assumption of a mass gap in the joint energy-momentum spectrum (i.e. an isolated, non-zero mass shell), otherwise we run into the notorious "infrared catastrophe", which is dealt with using "non-recoil" (i.e. Bloch-Nordsieck) approximation methods in formal perturbation theory but remains a challenge in a more rigorous setting, save in some non-relativistic models. In the second case, one is led to consider representations of the CCR's which are inequivalent to free field ones. Since field theories living in the whole space-time have infinite degrees of freedom, the Stone-von Neumann uniqueness theorem no longer holds (actually, Haag's theorem can be seen as a manifestation of this particular failure mechanism), and hence such representations should exist in abundance. Motivated by these results, Algebraic QFT was devised with a focus on structural (i.e. "model-independent") aspects of QFT in a way that does not depend on a particular representation; on other front, one may also try to explore this abundance of representations to construct models rigorously, which brings us to the realm of Constructive QFT.
Finally, it is important to notice that triviality of a model may stem from reasons unrelated to the underlying mechanism of Haag's theorem. The latter, once more, is a consequence of having an infinite number of degrees of freedom in infinite volumes (this theorem does not hold "in a box", for instance), whereas the former usually derives from an interaction which has too singular a short-distance behavior, as argued in the previous paragraph. This can be intuitively be understood by the (local) singularity and (global) integrability of the free field's Green functions: the lower the space-time dimension, the better the singular (UV) behaviour and the worse the integrability (IR) behaviour, and vice-versa. That's the underlying reason why $\lambda\phi^4$ scalar models are super-renormalizable in 2 and 3 dimensions (having only tadpole Feynman graphs as divergent in 2 dimensions) and non-perturbatively trivial in $>4$ dimensions.
Ah, I've almost forgotten about the references: in my opinion, the best discussion of triviality results in QFT from a rigorous viewpoint is the book by R. Fernández, J. Fröhlich and A. D. Sokal, "Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory" (Springer-Verlag, 1992), specially Chapter 13. There both the above "hard triviality" results for $\lambda\phi^4$ models and "soft triviality results" such as the Jost-Schroer-Pohlmeyer theorem (which underlies Haag's theorem, as mentioned at the beginning of my answer) are discussed. The book is not exactly for the faint of the heart, but the first sections of this Chapter provide a good discussion of the statements of the theorems, before proceeding to the proofs of the above "hard triviality" results. For a detailed discussion of Jost-Schroer-Pohlmeyer's and Haag's theorems, as well as their proofs, I recommend the book of J. T. Lopuszanski, "An Introduction to Symmetry and Supersymmetry in Quantum Field Theory" (World Scientific, 1991). The classic book of R. F. Streater and A. S. Wightman, "PCT, Spin and Statistics, and All That" (Princeton Univ. Press) also discusses these two results.