Preimage of a ball in a quotient metric space

Question

Let $(X,d)$ be a metric space and define for each $x\in X$ and for $\varepsilon >0$ the ball $B_\varepsilon^X (x) := \{ y:y\in X \land d(x,y) < \varepsilon\}$. Now, consider a subset $A\subseteq X$ and define the quotient space $X/_\sim$ as $X/A := \{[x]: x\in X\}$ and with equivalence relation $x\sim y \Longleftrightarrow (x=y) \lor (x,y\in A)$; furthermore consider the surjective quotient map (projection) $\pi: X \twoheadrightarrow X/A, x\mapsto [x]$.

Now the questions:

1. Is there any canonical or "forced" induced quotient metric on $X/_\sim$?
2. How does $B_\varepsilon^{X/_\sim} ([x]) := \{ [y]: [y]\in X/_\sim \land d([x],[y]) < \varepsilon \}$
3. How do I determine methodically $\pi^{-1}\big(B_\varepsilon^{X/_\sim} ([x])\big)$ for some $[x]\in X/_\sim$ ?