Etale coverings of certain open subschemes in Spec O_K

As Kevin points out, $V$ is indeed $\mathcal{O}_K[\frac{1}{2}]$ in your example. Your link to the fundamental group is also correct. $\pi_1(U)$ is the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside 2 (since you can restrict attention to the connected etale $\mathbb{Q}$-algebras)*. More generally, in your original question, these are replaced by $\mathcal{O}_L$ with the primes above the support of $D$ inverted, and the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside the support of $D$. These groups can get pretty horrendous, and so number-theorists tend to (at least in my view) study the more well-behaved but still very mysterious maximal pro-$p$-quotients of these groups. Now these pro-$p$ groups are not only finitely generated by work of Shafarevich ("Extensions with prescribed ramification points"), they are $d$-generated, where $d$ is the cardinality of the support of $D$ (for "tame" and not silly $D$). More impressively, the relation rank is also calculated/bounded (in this case, it's also $d$!!!), frequently leading to conclusions about when these groups are finite or infinite.

The "more complicated" starting point of $\mathcal{O}_K$ is in a sense not actually much more complicated. You're still asking for the Galois group of the maximal extension of $K$ unramified outside a finite set of primes, and Shafarevich's work gives formulas for the relation rank and generator rank for the pro-$p$-quotients.

The answer to your overall classification question is thus pretty difficult, and consume entire subdisciplines of number theory. As an example, Christian Maire has constructed number fields with a trivial class group but infinite unramified extensions -- you'd have to have a complete understanding of when this could happen before you could hope to prescribe all unramified extensions of a given degree, with or without certain primes inverted. There are certain cases where this can be done via, e.g., root discriminant bounds, but the story is far from being complete at this point.

As in Lars's answer, the situation is much much better understood for $\mathbb{P}_\mathbb{C}^1$ and Riemann surfaces than it is for number fields.

*: Actually, you have to be a little careful about 2-extensions. Etaleness doesn't pick up on whether or not infinite primes ramify. Everything's fine if you start with an totally imaginary field.

As to your addendum regarding the fundamental group of Riemann surfaces, the situation is as follows: If $X$ is a smooth projective algebraic curve of genus $g$ over an algebraically closed field $K$ of characteristic $0$, and if $U\subset X$ is obtained by removing $r$ distinct closed points, then $\pi_1(U)$ is the profinite completion of the surface group \[\left<a_1,\ldots, a_g, b_1,\ldots, b_g, c_1,\ldots, c_r | [a_1,b_1]\cdot\ldots\cdot[a_g,b_g]c_1\cdot\ldots\cdot c_r = 1\right>.\] Phrased more traditionally in terms of Galois theory of fields, if $K$ is the function field of $X$, then this group is precisely the Galois group of the maximal algebraic extension of $K$ which is unramified with respect to all valuations of $K$ except those corresponding to the $r$ points that were removed.

More concretely: The finite coverings of $U$ arise by taking finite extensions of $K$, unramified except possibly at the removed points, and taking the normalization of $U$ in $L$ (that is, $U$ is an affine curve given by a ring $A$ contained in $K$, the covering will be the spectrum of the normal closure of $A$ in $L$).

Edit: If $K$ is not algebraically closed and $U$ is geometrically connected, then there is a short exact sequence \[1\rightarrow \pi_1(U\times_K \bar{K}) \rightarrow \pi_1(U)\rightarrow Gal(\bar{K}/K)\rightarrow 1\] (one has to pick compatible base points), and if $X$ has a $K$-rational point, then this sequence splits, so the structure is "known", as far as the Galois group of the base is known.

Here is a result concerning Riemann surfaces which might be relevant (cf. your EDIT).

Fix a Riemann surface $X$, a discrete closed subset $D\subset X$ and put $X_0 =X\setminus D$ . Let $\mathcal RevRam (X;D)$ be the category with objects finite ramified coverings of $X$ (= proper non constant morphisms to $X$), étale over $X_0$. And let $\mathcal RevEt(X_0)$ be the category with objects finite étale coverings of $X_0$.

Theorem The restriction functor $\mathcal Revram (X;D) \to \mathcal Rev(X_0)$ is an equivalence of categories.

This is similar to Lars's interesting answer but the difference is that there is no assumption of algebraicity on the Riemann surfaces involved (for example $X$ could be an arbitrary open subset of $\mathbb C$)