Large cardinals

Which of these cardinals are regular? (Choose all that apply; assume ZFC.)

The cumulative hierarchy starts $V_0 = \varnothing$, $V_{n+1} = \mathcal P(V_n)$. How many elements does $V_4$ have?

Why is $\omega$ not measurable, even though non-principal ultrafilters on $\omega$ exist? (Consider countable completeness.)

In the proof that $V_\kappa \models \mathsf{ZFC}$ for inaccessible $\kappa$, which axiom's verification uses the regularity of $\kappa$ in an essential way?

Suppose ZFC proved "there is an inaccessible cardinal". What would follow?

Write IC for "there is an inaccessible cardinal". Which statements are correct? (Choose all that apply; assume ZFC + IC is consistent.)

Which Lean predicate correctly expresses that an ultrafilter U : Ultrafilter α is κ-complete?

In the proof that a measurable $\kappa$ is a strong limit, we take an injection $g : \kappa \to \mathcal P(\lambda)$ with $\lambda < \kappa$ and sets $Y_i \in U$ deciding whether $i \in g(\alpha)$. Why does $Y = \bigcap_{i < \lambda} Y_i$ have at most one element?

In Scott's proof that a measurable cardinal implies $V \ne L$, we assume $V = L$ and take $j : V \to M$ with critical point $\kappa$, the least measurable. Why is $M = V$?

Write laver_table(n) returning a dict rows with rows[p][q - 1] = p * q in the Laver table $A_n$ on $\{1, \dots, 2^n\}$: row $2^n$ is $1, 2, \dots, 2^n$, and for $p < 2^n$ fill the rows from $p = 2^n - 1$ down to $1$ using $p * 1 = p + 1$ and $p * (q + 1) = (p * q) * (p + 1)$. Then write period(row) returning the least period of a list.