I am trying to learn the lottery preparation technique from Hamkins' article but I am having some trouble understanding some of the technical details involved. I am at the beginning yet, so my difficulties are related with fast function forcing and the preservation of large cardinal notions when forcing with it. They are, probably, elemental difficulties arising from forcing but anyway I would like to ask here to get a deeper knowledge of this powerful technique.

For example, regarding Theorem 1.4 (preservation of weakly compactness), the author asserts that ``there are only $\kappa-$many dense sets for the forcing $\mathbb{F}_{\lambda,j(\kappa)}$'' so using enough closeness property of $M$ , and $<\kappa-closeness$ of $\mathbb{F}_{\lambda,j(\kappa)}$ (in fact, it is $\leq \lambda-$directed closed in $M$, though), we can get a $M-$generic $G$ in $V$. Why this quoted claim is true?

Also, regarding Theorem 1.5 (preservation of measurability), I find a similar trouble. Hamkins claims that ``since $2^\kappa=\kappa^+$, a simple counting argument shows that $|j(\kappa^+)|^V=\kappa^+$''. Why is this true and where it is used? I suspect that it is related to dense sets and the same argument as before applies to get a $M-$generic filter.

Every answer would be helpful. Thanks in advance!