Is it possible to reduce an NP language to a NEXP language with exponentially smaller input length?
This is quite unlikely to hold, because $\mathrm{EXP_{poly}^{NEXP}}$ actually coincides with $\Theta^{\exp}_2$, the exponential analogue of the class $\Theta^P_2$, which is presumably a strict subclass of $\mathrm{EXP^{NP}}$ (which is the exponential analogue of $\Delta^P_2$).
$\Theta^{\exp}_2$ can be variously defined as
$$\Theta^{\exp}_2=\mathrm{EXP^{\|NP}=EXP^{NP[poly]}=PSPACE^{NEXP}=P^{NEXP}=\exists\cdot DEXP},$$
where $\|$ denotes parallel (nonadaptive) access to the oracle, $\mathrm{[poly]}$ restricts the number of oracle queries to polynomial, the oracle tape is included in the space requirements of the $\mathrm{PSPACE}$ machine, and $\mathrm{DEXP}=\{L_0\smallsetminus L_1:L_0,L_1\in\mathrm{NEXP}\}$ is the exponential analogue of $\mathrm{DP}$.
For the $\mathrm{EXP_{poly}^{NEXP}}\subseteq\Theta^{\exp}_2$ inclusion, note that there are only exponentially many strings of polynomial length, hence the exponential-time machine may first ask all possible queries of that length in parallel, and then proceed with the computation, showing $\mathrm{EXP_{poly}^{NEXP}\subseteq EXP^{\|NP}}$.
For the $\Theta^{\exp}_2\subseteq\mathrm{EXP_{poly}^{NEXP}}$ inclusion, it is obvious that $\mathrm{P^{NEXP}\subseteq EXP_{poly}^{NEXP}}$.