Properties of Some Random Graphs

Yes, this model has been studied. You should look at Chapter 9 of Janson, Luczak and Rucinski's Random Graphs book, and in particular at Corollary 9.44. This corollary is in fact a rather well-known theorem, which I'll now explain.

Let $H_n(d)$ be the distribution you describe (Edit: more accurately, $H_n(d)$ is the distribution of the union of $d$ independent and uniformly random cycles, conditioned on the result being a simple graph), and let $G_n(2d)$ be the distribution of a uniformly random $2d$-regular (all nodes having degree exactly $2d$) simple graph. Then Corollary 9.44 states that for any fixed $d$, $H_n(d)$ and $G_n(2d)$ are contiguous, which means that for any graph property $A$,

$$ \mathbb{P}(H_n(d) \in A) \to 1~\mbox{as}~n\to\infty $$ if and only if $$\mathbb{P}(G_n(2d) \in A) \to 1~\mbox{as}~n\to\infty.$$

In other words, if you are only interested in studying whether things hold asymptotically almost surely, these two models are equivalent.

In particular, its isoperimetric constant is $(1/2+o(1)) d$, its diameter is $(1+o(1)) \log_{d-1} (n)$, and all eigenvalues except for the largest are $\sqrt{2(d-1)}+o(1)$.

For those interested in further reading about the contiguity of the above mentioned models for random regular graphs and generalization of the above: Catherine Greenhill, Svante Janson, Jeong Han Kim and Nicholas C. Wormald, Permutation pseudographs and contiguity, Combinatorics, Probability and Computing 11 (2002), 273 - 298.

Link: http://web.maths.unsw.edu.au/~csg/papers/gjkw-revised.pdf