What's the correlation between treewidth and instance hardness for random 3-SAT?
Not really an answer but the closest references I am aware of. There are results available for branch-width. Also, there is at least one empirical study of treewidth of industrial instances.


Discovering treewidth, H. Bodlaender, 2005. Gives algorithms to compute bounds on treewidth.
Treewidth in Industrial SAT Benchmarks, R. Mateescu, 2011. Uses variations of above algorithms to estimate treewidth of SAT2009 competition benchmarks.
Algorithms and complexity results for #SAT and Bayesian inference, F. Bacchus S. Dalmao, T. Pitassi, 2003. Relates branch-width to complexity of #SAT, 2003.
Satisfiability, branch-width and tseitin tautologies, Alekhnovich, Razborov, 2002.