Also, because a recent paper published in 2009 (hal.archives-ouvertes.fr/hal-00387303/document) that deals with this question has only 3 citations on google scalar (none of which seem to solve this problem), and also because the handbook published in 2011 doesn't have a complete solution to this seemingly natural question, it's likely still open, although you should ask a graph theorist to be sure.

I'm no graph theorist. And probably you already know this, but Exercises 5.15-5.17 on pages 59 and 60 of the Handbook of Product Graphs (2nd edition) seem to characterize such $G$ and $H$ for some special cases.

Do you mean $(u,v) \ E \ (u',v') \Leftrightarrow u E_1 u' \wedge v E_2 v'$ (or, equivalently, $v$ and $u$ in $(u,v) \ E \ (u',v') \Leftrightarrow u E_1 u' \wedge v E_2 v'$)?

Ok. I will write up something with a little more detail like the Schönheim bound.

The smallest size of such $\mathcal{F}$ is called the covering number $C_1(n,k,t)$. I think this is the website The Masked Avenger mentioned: ccrwest.org/cover.html

@HaoChen Wow, thanks. You shouldn't have.

@HaoChen Thanks for the info. It was a bit of a surprise to me that there are so few papers on this topic. Actually, I need an LDPC code version of mixed codes for my research now, and all I could find were a couple results... Oh, and I just noticed you edited your answer before I gave the link to the paper by Perkins, Sakhonivich, and Smith. Have my upvote for beating me to it!