I don't think that
torsion in the homology has been ruled out
Certainly, torsion in Cech cohomology has been ruled out for a compact subset. The "usual" universal coefficient formula, relating Cech cohomology to $\operatorname{Hom}$ and $\operatorname{Ext}$ of Steenrod homology, is not valid for arbitrary compact subsets of $\Bbb R^3$ (although it is valid for ANRs, possibly non-compact). The "reversed" universal coefficient formula, relating Steenrod homology to $\operatorname{Hom}$ and $\operatorname{Ext}$ of Cech cohomology is valid for compact metric spaces, but it does not help, because $\operatorname{Ext}(\Bbb Z[\frac1p],\Bbb Z)\simeq\Bbb Z_p/\Bbb Z\supset\Bbb Z_{(p)}/\Bbb Z$, which contains $q$-torsion for all primes $q\ne p$. (Here $\Bbb Z_{(p)}$ denotes the localization at the prime $p$, and $\Bbb Z_p$ denotes the $p$-adic integers.
The two UCFs can be found in Bredon's Sheaf Theory, 2nd edition, equation (9) on p.292
in Section V.3 and Theorem V.12.8.)
The remark on $\operatorname{Ext}$ can be made into an actual example. The $p$-adic solenoid $\Sigma$ is a subset of $\Bbb R^3$. The zeroth Steenrod homology $H_0(\Sigma)$ is isomorphic by the Alexander duality to $H^2(\Bbb R^3\setminus\Sigma)$. This is a cohomology group of an open $3$-manifold contained in $\Bbb R^3$, yet it is isomorphic to $\Bbb Z\oplus(\Bbb Z_p/\Bbb Z)$ (using the UCF, or the Milnor short exact sequence with $\lim^1$), which contains torsion. Of course, every cocycle representing torsion is "vanishing", i.e. its restriction to each compact submanifold is null-cohomologous within that submanifold.
By similar arguments, $H_i(X)$ (Steenrod homology) contains no torsion for $i>0$ for every compact subset $X$ of $\Bbb R^3$.
It is obvious that "Cech homology" contains no torsion (even for a noncompact subset $X$ of $\Bbb R^3$), because it is the inverse limit of the homology groups of polyhedral neighborhoods of $X$ in $\Bbb R^3$. But I don't think this is to be taken seriously, because "Cech homology" is not a homology theory (it does not satisfy the exact sequence of pair). The homology theory corresponding to Cech cohomology is Steenrod homology (which consists of "Cech homology" plus a $\lim^1$-correction term). Some references for Steenrod homology are Steenrod's original paper in Ann. Math. (1940), Milnor's 1961 preprint (published in http://www.maths.ed.ac.uk/~aar/books/novikov1.pdf), Massey's book Homology and Cohomology Theory. An Approach Based on Alexander-Spanier Cochains, Bredon's book Sheaf Theory (as long as the sheaf is constant and has finitely generated stalks) and the paper
As for torsion in singular $4$-homology of the Barratt-Milnor example, this is really a question about framed surface links in $S^4$ (see the proof of theorem 1.1 in the linked paper).
Hi Tony.
This is not really a homology-question, the core of it is the fundamental group. The homomorphism you are using is used in the study of Van Kampen diagrams. Consider a presentation $G=\langle A|R\rangle$. A Van Kampen diagram on $S$ is a labeled graph like you have defined it. The only difference is that in a Van Kampen diagram all labels are generators (or their inverses) $a^{\pm 1}$ and not arbitrary words (although you could define it in this general way without problems because of the Van Kampen lemma).
Then every path in this graph has a group word written on it and "reading the word along a path" is a homomorphism {Paths}$\to G$ with respect to composition of paths. It turns out, that this is compatible with homotopy of paths so this induces a homomorphism $\pi_1(S,x_0)\to G$.
This is the general version of your homomorphism: If your $G$ happens to be abelian, then this homomorphism factorizes through $\pi_1(S,x_0)^{ab}$ which is $H_1(S)$ by the Hurewicz theorem.
This point of view clarifies some connections between the geometry of Van Kampen diagrams and group theoretic questions.
For example the Van Kampen lemma tells you that a group word is trivial if and only if there is a Van Kampen diagram on this disk with this word written on the boundary.
Another fact is this one: If there are no nontrivial "reduced" Van Kampen diagrams on the torus, then every two commuting elements of $G$ generate a cyclic subgroup (i.e. $xyx^{-1}y^{-1}=1$ has only the trivial solutions $x=a^k, y=a^m$ for some $a\in G$.). In a similar spirit one can prove: If there are no nontrivial reduced Van Kampen diagrams on the real projective plane, then there are no involutions in $G$ (i.e. $x^2=1$ has only the trivial solution $x=1$), and if there are no nontrivial reduced Van Kampen diagrams on Klein's bottle, then the only element that is conjugated to its own inverse is the identity (i.e. $yxy^{-1}=x^{-1}$ has only the trivial solution $x=1$).
This connection between geometry and group properties becomes less obscure, if one knows the fundamental groups of the disk (1), the torus ($\langle x,y | xyx^{-1}y^{-1}=1\rangle$), the real projective plane ($\langle x | x^2=1\rangle$) and Klein's bottle ($\langle x,y | yxy^{-1}=x^{-1}\rangle$).
Best Answer
Expanding on Ryan's comment, one way to get the torsion linking form you mention (at least for compact oriented manifolds) is through the sequence of isomorphisms $$Tor(H_l(X;\mathbb{Z}))\cong Tor(H^{n-l}(X;\mathbb{Z})) \cong Ext(H_{n-l-1}(X);\mathbb{Z})\cong Ext(Tor(H_{n-l-1}(X));\mathbb{Z})\cong Hom(Tor(H_{n-l-1}(X));\mathbb{Q}/\mathbb{Z}) .$$ The first isomorphism is Poincar'e duality, and the second is the universal coefficient theorem (using that the Ext summand corresponds to the torsion subgroup). The third is a basic property of Ext (write the homology group as a sum of a torsion group and a free group - this uses that the homology is finitely generated from the compactness assumption). The final can be seen to come from the short exact sequence you mention: take the Hom/Ext exact sequence and note that $Ext(-,\mathbb{Q})=0$ always and $Hom(A,\mathbb{Z})=0$ when $A$ is torsion.
So all this can be generalized in circumstances in which these properties all hold. For example, if $R$ is a Dedekind ring and $X$ is a compact $R$-orientable manifold, I believe everything should go through to give a pairing $Tor(H_l(X;R))\otimes Tor(H_{n-l-1}(X;R))\to Q(R)/R$, where $Q(R)$ is the field of quotients of $R$.
Offhand, I'm not sure about a version in which the coefficients are modules over $R$ (in your original question, modules over $\mathbb{Z}$). One would have to think through in what cases versions of these isomorphisms still hold.
*Note: You don't mention manifolds explicitly, but the site you link to for your basic definitions does assume the context of closed, oriented manifolds.
**It does take a little work to show that this definition of the linking form is equivalent to the one using the Bockstein.