Difference between $NP\cap coNP$ and $NEXP\cap coNEXP$
NO. Assume to the contrary that we have that equality, the LHS contains $EXP^{EXP}=EEXP$. So, we deduce that $NEXP=EEXP$.But then the LHS can even contain $EXP^{EEXP}=EEEXP$. That is way too much over $NEXP$.
YES. By the same proof for $P^{NP\cap coNP}=NP\cap coNP$, the NEXP machine simulates the oracle machine by guessing the oracle answer and the corresponding certificate of membership (non-membership) and verify by the verifier (refuter, resp.) of the NEXP (coNEXP, resp.) machine for the language of the oracle. Then, if at any time the certuficate does not prove the guess, refute and halt. Otherwise, return the same answer as the oracle machine.


Similarly for the coNEXP machine, but you have to accept if a guess is not proven by the certificate guessed after it.