Is $x \mapsto d(x, A)$ a quotient map?

This can fail if $X$ is not connected but the image of $d_A$ is. For example consider $X = (-1, 0] \cup [1, \infty)$ and $A = \{0\}$, with the Euclidean metric $d_A(x) = | x |$. Then $d_A(X) = [0, \infty)$, which has one component, but no pair of points in distinct components of $X$ has been identified.

In the case $X$ is connected, I don’t see an obvious answer.

For your second question. $A$ is compact doesn't imply $d_A$ is a closed map. Let $X$ be the space of irrational numbers and $A$ be $\sqrt 2$. Then $B=[0,\sqrt 2]\cap X$ is closed in $X$. But $\sqrt 2\notin d_A (B),$ although it belongs to $d_A(X)$. It's easy to see $\sqrt 2\in \overline{d_A(B)}$, hence $d_A(B)$ is not closed.