Sorry (i) has big error. dim=q is not justified yet, my mistake.

(i) Under the assumption gcd$(d,...)$, $\beta_1^d\neq \beta_2^d$ so that the subspace has dimension at least 2. It should be then $q$. (ii) In the first case, (if I'm correct) $d$-power preserves basis property, at each $B_j$'s. Consequently, a basis of $\mathbb{F}_{p^k}$ is replaced by another basis by the $d$-power operation : the product of new $\mathcal{B}$'s gives a new basis of $\mathbb{F}_{p^k}$. Thus $M_{p^k}^{(d)}=k$.