Two observations (with $k$ a field of char. zero):

• If $A$ is a domain over a field $k$, then elements of $D(A)$ extend to elements of $D(A_S)$ for all mult.closed sets $S$.

• Your questions become easier if you ask about the algebra $\Delta(A)$ of $D(A)$ generated by $A$ and derivations: then the answer is yes to your two questions. If $A$ is finitely generated and regular, then $D(A)=\Delta(A)$.

You'll find this in the last chapter of McConnell and Robson's book on noetherian rings.

(Finally, $D^{\leq n}(A)$ is an $A$-bimodule which is not symmetric, so tensoring with $A_f$ on one side or the other is not the same thing)

Two observations (with $k$ a field of characteristic zero):

• If $A$ is a domain over a field $k$, then elements of $D(A)$ extend to elements of $D(A_S)$ for all multiplicatively closed sets $S$.

• Your questions become easier if you ask instead about the subalgebra $\Delta(A)$ of $D(A)$ generated by $A$ and derivations: then the answer is yes to your two questions. Now, if $A$ is finitely generated and regular, then $D(A)=\Delta(A)$, so in this case the answer is yes for $D(A)$ too.

You'll find this in the last chapter of McConnell and Robson's book on noetherian rings.

(Finally, $D^{\leq n}(A)$ is an $A$-bimodule which is not symmetric, so tensoring with $A_f$ on one side or the other is not the same thing)