The answer is yes, but only after tensoring with $\mathbb R$.
Thinking of the Beilinson regulator map with values in Deligne cohomology is simpler than thinking about the Borel regulator map; it's been proved that they agree with each other.
The topological Chern character map $ch_n : kU_{i} \to H^{2n-i}(pt,\mathbb Q)$ is an isomorphism $kU_{2n} \otimes \mathbb Q \ \xrightarrow \cong \ \mathbb Q$ when $i=2n$.
The corresponding algebraic Chern character map with values on Deligne cohomology is $ch_n : K_{i}\mathbb C \to H^{2n-i}_{\cal D}(pt,\mathbb Q(n))$. Here $\mathbb Q(n)$ (or $\mathbb Z(n)$) denotes a certain cochain complex of sheaves for the analytic topology on a complex manifold $X$. It starts in degree 0 with $\mathbb Q$ (resp., $\mathbb Z$), in cohomological degree 1 it has $\mathbb C$, and the differential map $d^0 : \mathbb Q \to \mathbb C$ is multiplication by $(2 \pi i)^ n$. The term in degree $i+1$ is the sheaf of holomorphic differentials $\Omega^i$ if $i < n$ and is $0$ if $i \ge n$. The exponential map $\mathbb C \to \mathbb C ^ \times $ given by $z \mapsto e^z$ gives a quasi-isomorphism $ \mathbb Z (1) \to \mathbb C ^ \times [-1]$; the degree shift there answers your second question, partially; another way of saying that is that there is a degree shift in the boundary map $c_1 : H^1(X,\mathbb C^\times) \to H^2(X,\mathbb Z)$. I say "partially", because one must know also that the regulator map involves no further degree shift; in degree 1 it's because the map $\mathbb C ^ \times \to \mathbb R$ given by $z \mapsto {\rm log} |z|$ involves no degree shift.
Now consider the projection $\mathbb C \to \mathbb R$ that sends $ (2 \pi i)^n $ to $0$ and $i^{n-1}$ to $1$; perhaps there is a better normalization for this map, such as choosing to send $(2 \pi i)^{n-1}$ to $1$. It induces a map of cochain complexes $\mathbb Q(n) \to \mathbb R[-1]$; the map it induces on Deligne cohomology, composed with the Chern character map above, is the Beilinson regulator map $$ch_n : K_{i}\mathbb C \to H^{2n-i}(pt,\mathbb R[-1]) = H^{2n-i-1}(pt,\mathbb R),$$whose only nonzero possibility is the map $ch_{n} : K_{2n-1}\mathbb C \to H^{0}(pt,\mathbb R) = \mathbb R$. For the ring of integers $A$, we get a map $K_{2n-1} A \to K_{2n-1}( A \otimes \mathbb C ) \to H^{0}(Spec(A \otimes \mathbb C),\mathbb R) = \mathbb R^{s+2t}$. Borel's theorem is recast as saying that for $n > 1$ the map induces an isomorphism of $(K_{2n-1} A) \otimes \mathbb R$ with the appropriate eigenspace for the action of $G = Gal(\mathbb C/\mathbb R)$ on $$H^0(Spec(A \otimes \mathbb C), \mathbb R (n-1)) = \mathbb R^{s+2t},$$where now I use $\mathbb R (n-1)$ to remind us how $G$ acts on this real vector space of dimension 1.
(Added later: actually, it may be more natural to replace the $\Sigma$ in the question by $\Omega$ and to use the anti-invariants (or invariants, depending on the parity of n) under $G$ acting on $K^{top}(A \otimes \mathbb C) \otimes \mathbb C$ instead of the invariants. Thus the degree shift can be viewed as $-1 = 1 - 2$)
Best Answer
I know nothing about Alexander polynomials but let me try to answer the Iwasawa theory part. As is well known, in classical Iwasawa theory one considers cyclotomic $\mathbb{Z}_p$ extension $F{\infty}$ of $F$. We take the $p$-part of the ideal class group $A_n$ of the intermediate extension $F_n$ of $F$ of degree $p^n$. The inverse limit of $A_n$ with respect to norm maps, say $A$, has an action of $G= Gal(F_{\infty}/F)$. Since $A$ is pro-$p$, it becomes a $\mathbb{Z}_p[G]$-module. However, it is not finitely generated over this group ring (and for various other reasons) one considers the completion $\mathbb{Z}_p[[G]]$ of $\mathbb{Z}_p[G]$. Since $A$ is compact, it becomes a $\mathbb{Z}_p[[G]]$-module. As $G \cong \mathbb{Z}_p$, the ring $\mathbb{Z}_p[[G]] \cong \mathbb{Z}_p[[T]]$, the power series ring in variable $T$. There is a nice structure theory for finitely generated modules over $\mathbb{Z}_p[[T]]$. The module $A$ is a torsion $\mathbb{Z}_p[[G]]$-module (i.e.$Frac(\mathbb{Z}_p[[G]]) \otimes A = 0$). For such modules one can define the characteristic ideal using the structure theory. Iwasawa's main conjecture asserts that there is a canonical generator for this ideal called the $p$-adic $L$-function.
In generalised Iwasawa theory (more precisely, to formulate the generalised main conjecture à la Kato), one wants to consider extensions whose Galois groups are not necessarily $\mathbb{Z}_p$ (but most formulations of the main conjecture still require that the cyclotomic $\mathbb{Z}_p$-extension of the base field be in the extension). For the completed $p$-adic groups rings of such Galois groups, the structure theory completely breaks down even if the Galois group is abelian.
However, one can still show that $A$ is a torsion Iwasawa module (which again just means that $Frac(\mathbb{Z}_p[[G]]) \otimes A = 0$. Note that it is always possible to invert all non-zero divisors in a ring even in the non-commutative setting). Hence the class of $A$ in the group $K_0(\mathbb{Z}_p[[G]])$ is zero. Strictly speaking, here I must assume that $G$ has no $p$-torsion so that I can take a finite projective resolution of $A$, or I must work with complexes whose cohomologies are closely related to $A$. But I will sinfully ignore this technicality here. Now, since the class of $A$ in $K_0(\mathbb{Z}_p[[G]])$ is zero, there is a path from $A$ to the trivial module 0 in the $K$-theory space. In Iwasawa theory this is most commonly written as
There exists an isomorphism $Det_{\mathbb{Z}_p[[G]]}(A)$ $\to$ $Det(0)$.
This isomorphism replaces the characteristic ideal used in the classical Iwasawa theory. The $p$-adic $L$-function then is a special isomorphism of this kind. (Well one has to be careful about the uniqueness statement in the noncommutative setting but it is a reasonably canonical isomorphism). Hence the main conjecture now just asserts existence of such a $p$-adic $L$-function.
Thus the $p$-adic $L$-function may be thought of as a canonical path in the $K$-theory space joining the image of Selmer module (or better- a Selmer complex), such as the ideal class group in the above example, and the image of the trivial module.
I hope this answer helps until Minhyong sheds more light on his remarks and relations between $p$-adic $L$-functions and the Alexander polynomials.
[EDIT: Sep. 30th] To answer Daniel's questions below- 1) Take the projective resolution of A to define its class in $K_0$. In any case the $K$-theory of the category of finitely generated modules is the same as the $K$-theory of finite generated projective modules. 2) I do not know if there are any sensible/canonical multiplicative sets at which to localise $\mathbb{Z}_p[G]$. The modules considered in Iwasawa theory are usually compact (or co-compact depending on whether you take inverse limit with respect to norms or direct limit with respect to inclusion of fields in a tower) and so the action of the ring $\mathbb{Z}_p[G]$ extends to an action of the completion $\mathbb{Z}_p[[G]]$ by which we mean $\varprojlim \mathbb{Z}_p[G/U]$, where $U$ runs through open normal subgroups of $G$. 3) We do not usually get a loop in the $K$-theory space of $\mathbb{Z}_p[[G]]$. However, modules which come up from arithmetic are usually torsion as $\mathbb{Z}_p[[G]]$-modules i.e. every element in the module is annihilated by a non-zero divisor or in other words if $X$ is the module then $Frac(\mathbb{Z}_p[[G]]) \otimes X=0$. (it is usually very hard to prove that the modules arising are in fact torsion). But once we know that they are torsion we get a loop in the $K$-theory space of $Frac(\mathbb{Z}_p[[G]])$. If we know that for some multiplicatively closed subset $S$ of $\mathbb{Z}_p[[T]]$ annihilates $X$ i.e. $(\mathbb{Z}_p[[G]]_S) \otimes X=0$, then we already get a loop in the $K$-theory space of $\mathbb{Z}_p[[G]]$. In noncommutative Iwasawa theory it is often necessary to work with a multiplicative set strictly smaller than all non-zero divisors.