Given a computable function $f$ and a decidable language $L$, is there a Turing machine $M$ such that $M$ both decides $L$ and computes $f$?
Yes it clearly exists; informally you can build such $M$ in this way:


$M$ makes a copy of its input $x$
then without touching such a copy, $M$ decides if $x$ is in $L$ (possibly "destroying" the working copy of $x$), 
if $x \in L$ it enters a "subprogram" (set of states) in which all states are accepting and such subprogram uses the copy of $x$ to compute $f(x)$ (leaving the result in the output tape and ending in an accepting state)
if $x \notin L$ it enters a "subprogram" (set of states) in which all states are NON accepting and such subprogram uses the copy of $x$ to compute $f(x)$ (leaving the result in the output tape and ending in a NON accepting state)