upper bound on the size of a DFA for A|B given the DFAs for A and B?
The upper bound of $nm$ follows from the usual Cartesian product construction that you will find in just about any book on automata theory.  A lower bound of $nm$ is observed (for the case $\operatorname{gcd}(n,m) = 1$) in this paper:


  Sheng Yu.  State Complexity of Regular Languages.  Journal of Automata, Languages and Combinatorics, 2001.


The lower bound is proved for DFAs that I would consider to exist in the real world -- but if you want an expectation you will have to be more precise about the distribution from which $A$ and $B$ are drawn.

The bound is simple and can surely be extended to three DFAs.