Question about Silver forcing theorem proof

Question

I'm trying to understand the proof of the following Theorem:

If there is a supercompact cardinal $\kappa$, then there exists a generic extension where $\kappa$ is a measurable cardinal and $2^\kappa > \kappa^+$.

The proof is exhibited in Jech's Set Theory (3rd Millennium edition) (p.397, Proof of Theorem 21.4).

My questions:

1. Right at the beginning of the proof we assume that $2^\kappa = \kappa^+$. Why can we do this?
2. I do not fully get the proof idea. The idea is that one employs iterated forcing with Easton support, where to each inaccessible cardinal $\alpha \leq \kappa$ one successively adjoins $\alpha^{++}$ subsets of $\alpha$. Why designate the inaccessible cardinals?
3. I'm not really sure, how it was managed to prove that $Q_\kappa$ is a forcing notion that adjoins $\kappa^{++}$ subsets of $\kappa$ and is cardinal preserving.
4. The final part of the proof is about measurability. We first construct an elementary embedding of $V[G]$ which we obtain by supercompactness of $\kappa$. Why is $|P| = \kappa^{++}$? Why is $P \in \mathcal{M}$?

Please excuse if some of these questions are very simple to you. I would like to understand this theorem.