Is $\bigcup_{c \ge 1} \mathsf{DTime}(2^{cn})$ closed under polynomial reduction?
No, this would contradict the time hierarchy theorem. Note that any language in $\mathsf{EXP}$ is polynomial time reducible to some language in $\mathsf{E}$ by padding. Suppose $L\in \mathsf{DTIME\big(2^{n^c}\big)}$, then consider $L'=\big\{x0^{|x|^c}\big|x \in L\big\}$. Obviously $L'\in \mathsf{E}$ and $L\le_p L'$, thus if $\mathsf{E}$ is closed under polynomial time reductions we have $L\in\mathsf{E}$ and consequently $\mathsf{E=EXP}$.