Independent statements about generated $\sigma$-algebras

Question

I base my question on this. I saw it at Math Overflow and as nobody answer it, although its bounty, I decided to post it here.

Consider the statements $P$ and $Q$:

$P$: Every subset of the plane belongs to the $\sigma$-algebra generated by arbitrary rectangle.

That is, $C\subseteq \Bbb{R}^2 \implies C\in \sigma(\{A\times B : A\subseteq \Bbb{R} \mbox{ and } B\subseteq \Bbb{R}\})$. And

$Q$: Every continuum-sized family of subsets of $\mathbb{R}$ is contained in a countably generated $\sigma$-algebra.

Here is proved that statement $P$ is independent from $ZFC$; and here is (partially) proved that $Q$ is also independent from $ZFC$.

Question 1: Why does $P\implies Q$ ? As the author of the initial question said.

Question 2: Does $Q\implies P$ ?

I think Question 1 must be much simpler, but I'm just curious about this and I'm far from being a specialist at the matter. I'm sorry for the lack of development of a try :(

Thank you for any help

Answer 1

First part of this must help you on $P\implies Q$