The answer is positive.

For each $n$ we can find effectively an index $s(n)>n$ such that $Range(\phi_{s(n)})=Domain(\phi_n)$. Then $Range(s)$ is computable, and we can define $$ \psi_x(y)=\begin{cases}\phi_{n}(y), & \mbox{if }x=s(n);\cr 0,& \mbox{if }x\notin Range(s)\ \& \ y\in Range(\phi_x);\cr \uparrow&\mbox{otherwise}. \end{cases} $$ Now $\phi_n=\psi_{s(n)}$ for all $n$, and $Range(\phi_x)=Domain(\psi_x)$ for all $x$.

The answer is positive.

For each $n$ we can find effectively an index $s(n)$ such that $Range(\phi_{s(n)})=Domain(\phi_n)$ and $s(n)>s(m)$ for all $m<n$. Then $Range(s)$ is computable, and we can define $$ \psi_x(y)=\begin{cases}\phi_{n}(y), & \mbox{if }x=s(n);\cr 0,& \mbox{if }x\notin Range(s)\ \& \ y\in Range(\phi_x);\cr \uparrow&\mbox{otherwise}. \end{cases} $$ Now $\phi_n=\psi_{s(n)}$ for all $n$, and $Range(\phi_x)=Domain(\psi_x)$ for all $x$.