Algorithms for graph generation given parameters
This question was posted more than 8 years ago, and much progress was done since then. However, many questions remain open, even very natural ones like sampling a random graph with prescribed clustering coefficient. Also, sampling simple graphs (no loop, no multi-edge) with prescribed degree sequence made much progress but remains difficult.
I would like to give a pointer to a quite comprehensive and recent survey on many aspects of these questions, which I think may be of interest to most readers of this post:
Recent Advances in Scalable Network Generation by
Manuel Penschuck, Ulrik Brandes, Michael Hamann, Sebastian Lamm, Ulrich Meyer, Ilya Safro, Peter Sanders, Christian Schulz, 2020
For the specific problem of graphs with prescribed degree sequence, I recommend:

Configuring Random Graph Models with Fixed Degree Sequences by
Bailey K. Fosdick, Daniel B. Larremore, Joel Nishimura, Johan Ugander, 2016
Fast uniform generation of random graphs with given degree sequences by
Andrii Arman, Pu Gao, Nicholas Wormald, 2020

but this is a very partial list, of course.
The last reference has a dedicated discussion of the regular case.