The answer to question 2 is yes. To see this, it suffices to produce from any well-ordering of some $V_\alpha$ a set of ordinals, such that the well-ordering can be reconstructed from the set of ordinals. Given a well-ordering of $V_\alpha$, this ordering has some length $\kappa$, and so there is a relation $E$ on $\kappa$ and an isomorphism $\langle V_\alpha,\in\rangle\cong\langle\kappa,E\langle$$\langle V_\alpha,\in\rangle\cong\langle\kappa,E\rangle$, which is just the bijection determined by the order-isomorphism between $\leq$ on $V_\alpha$ and the natural order on $\kappa$. The relation $E$ is a set of pairs of ordinals, and we may by the usual pairing function code all the information of $\langle\kappa,E\rangle$ by a set of ordinals $A$. From $A$ we may recover $E$ and therefore (by the Mostowski collapse) the set $V_\alpha$ and by the bijection to $\kappa$ we recover the order $\leq$. So this provides a (definable) injective map from $W$ to $P(\text{Ord})$, as desired.
It follows that the answer to question 3 is also affirmative, since from any injection from $W$ to $P(\text{Ord})$ we get a surjection in the other direction.