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You have given that the equivalence class of $x$ and the equivalence class of $y$ is equal. $[x]=[y]$. Does this imply that $x\sim y$? If yes, how do I prove it, if no, what sort of counter-examples can I give?

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By definiton of the equivalence class of some element, we have \[ [x] = \{z \in X \mid z \sim x\} \] By reflexivity of $\sim$, we have $x \in [x]$. So $y \in [y] = [x]$, that is $y \in [x]$, so by definition of $[x]$, $y \sim x$.

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Yes. If $[a]$ and $[b]$ are equivalence classes, then either $[a]=[b]$ or $[a]\cap[b] = \varnothing$.

Equivalence classes always form a partition of the underlying set: every element $x$ lies in exactly one equivalence class.

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