Do Gap-CDH groups exist? A Gap-CDH group is such that, given group elements $g, a = g^x, b = g^y$, it is hard to compute $g^{xy}$, but, given a group element $c$, easy to verify if $c = g^{xy}$. While such groups have been proposed, and it is obviously feasible to define a group where Decision Diffie-Hellman is demonstratively easy, exactly what grounds exist for the conjecture that Computational Diffie-Hellman is hard in such groups, beyond the observation that no known algorithms for solving it yet exist? 

Are there ways to reduce the hardness of CDH in a Gap-CDH group (in general or some specific group), to the hardness of CDH in a group that has received more thorough scrutiny over the years? For instance, does there exist a group $G$, such that DDH is easy in $G$, and there exists a proof that the ability to solve CDH in $G$ implies the ability to solve CDH in $Z_p^*$ for a suitably large prime $p$?
The closest that I can think of is a "symmetric bilinear group" (a.k.a. Type-I pairing group) that was popular when bilinear groups were first introduced. This is actually a pair of groups $(G, G_T)$ together with an efficient non-degenerate bilinear map $\otimes: G \times G \to G_T$. Obviously DDH is easy in $G$, since one can on input $(g, U, V, W) \in (G_1)^3$ test whether $g \otimes W = U \otimes V$. But the pairing will not necessarily help you compute CDH in the source group, since the map $(\cdot \otimes g)$ is not meant to be easy to invert.

These kinds of groups are constructed based on various types of elliptic curve, my intuition is that a pairing "out of" a curve group won't help you compute new elements in the group itself.

This kind of group appeared in Boneh, Boyen and Shacham (eprint 2004/174) where they used the Decision Linear assumption instead of DDH, since DLIN can be hard even in the presence of a pairing.