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.