Spectral radius of the product of a block-diagonal matrix and a row stochastic matrix

Question

Let $W\in\mathbb{R}^{N\times N}$ be a right (row) stochastic matrix with non-negative $ij$ entries $w_{ij}\geq0$, where $\sum_{j=1}^N w_{ij} = 1$, and let $A\in\mathbb{R}^{nN\times nN}$ be a block-diagonal matrix with blocks $A_i \in \mathbb{R}^{n\times n}$.

I am trying to figure out if there is a way of studying the spectral radius of the matrix $$ B=A(W\otimes I_n),$$ where $\otimes$ is the Kronecker product and $I_n$ is the $n\times n$ identity matrix. Specifically, there are two questions that I am trying to answer:

1. Given the eigenvalues of each block $A_i$, can we design $W$ such that $\rho(B)<1$?
2. Given the matrix $W$, can we specify the eigenvalues of the blocks $A_i$ so that $\rho(B)<1$?

Please forgive my ignorance!