As Kevin points out, $V$ is indeed $\mathcal{O}_K[1/2]$ 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. More generally, in your original question, these are replaced by $\mathcal{O}_K$ 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. These pro-$p$ groups are finitely generated by work of Shafarevich ("Extensions with prescribed ramification points") -- in fact, they are $d$-generated, where $d$ is the cardinality of the support of $D$ (for reasonable $D$). 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 not actually much more complicated (in some sense). 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.
Ack, have to run. I'm going to come back and elaborate (and possibly proof-read) in the near future.