On complete embeddings between forcing notions

Question

I was wondering whether the following two statement are true and why:

1. For any two forcing notions $\mathbb{P,Q}$, if $\ \Vdash_\mathbb{Q} ``$There is a $\check{\mathbb{P}}$-generic filter over $\dot{V}"$, where $\dot{V}$ is the canonical name for the ground model, then there is a complete embedding from $\mathbb{P}$ to $\mathbb{Q}$.
2. For any two forcing notions $\mathbb{P,Q}$, if there is a complete embedding from $\mathbb{P\times P}$ to $\mathbb{P\times Q}$ then there is also one from $\mathbb{P}$ to $\mathbb{Q}$

Both statements seems relatively simple, but nonetheless I'm having difficulties in proving them (or disproving).
Any hint or reference is much appreciated.

Answer 1

These are completely false.

For the first one, let $\Bbb Q$ be the Cohen forcing, and let $\Bbb P$ be the lottery sum of uncountably many Cohen forcings. Namely, $\Bbb P$ is an iteration of "pick a countable ordinal, then add a Cohen real".

In the second case, let $\Bbb P$ be the collapse $\operatorname{Col}(\omega,\omega_1)$, and let $\Bbb Q$ be the Cohen forcing.