is co-NP-hard contained in EXPTIME-hard or vice-versa?
It is known that co-NP $\subseteq$ PSPACE $\subseteq$ EXPTIME, so every EXPTIME-hard problem, must also be a co-NP-hard problem (as there is a polynomial time reduction from every $L\in$ EXPTIME to it, and therefore also from every $L\in$ co-NP)
Therefore EXPTIME-hard $\subseteq$ co-NP-hard.
Note that this argument does not hold for co-NP-complete and EXPTIME-complete. This is still an open problem (i believe).