Is there (I strongly presume not but not seeing how to show it) a (non-zero) arithmetic degree $a$ such that for all arithmetic degrees $e \not\geq_a a$ we have $e \oplus a$ is a minimal cover of $e$ in the arithmetic degrees (i.e. there is no degree $c$ with $e <_a c <_a e \oplus a$).

In the Turing degrees I'd use a variant of the Posner-Robinson join theorem to build $E$ with $E' \equiv_T E \oplus A$.  However, in the paper "REA operators, R.E. degrees and Minimal Covers" by Jockusch and Shore it's listed as an open question whether or not for all non-arithmetic $X$ there is a $Y$ with $Y^{(\omega)} \leq_T T \oplus X$ (and I'm unsure about the arithmetic version).  For that matter, I'm not even sure that $Y^{(\omega)}$ isn't a minimal cover in the arithmetic degrees (I'll ask in a separate question).