I'm confused. I would have expected $\textrm{Coh} \mathbb P^2$ to have a Grothendieck group of rank 3, which is inconsistent with what is being claimed here. Could you please clarify?

Sorry, missed that. Since some of the boundary of $P_1$ is contained in $P_2$ and some of it isn't, there must be some intersection between the facets of $P_1$ and $P_2$. But really, you would do better with this question at math.stackexchange.com.

No, the facets needn't intersect. It's simpler just to think inside the $n-1$-dimensional hyperplane. There, $P_2$ is full-dimensional, so we could think of it as being quite large, with $P_1$ entirely contained inside it. But you would probably do better with this question on math.stackexchange.com.

Let me note that the case $m=2$ is solved in the linked question.

That seems way too big to me, but it's very possible I'm missing something. Should the bound of $2^{\omega(d)}$ be obvious to me?