Let $M_{2^k}=M_2\otimes M_2\otimes\cdots \otimes M_2$ and $P_{2^k}=P_2\otimes P_2\otimes\cdots \otimes P_2$ where $M_2=\left[\begin{matrix} 1&1\\ 1&0\\ \end{matrix}\right]$ and $P_2=\left[\begin{matrix} \frac{1}{4}&\frac{1}{4}\\ \frac{1}{4}&\frac{1}{4}\\ \end{matrix}\right]$ and left $F=(-1)^{\circ M}$ where $\circ$ refers to element wise power operation.
Let $x,y\in \{0,1\}^{2^k}$.
Is $F_{2^k}\odot P_{2^k}= (F_2\odot P_2)\otimes(F_2\odot P_2)\otimes\cdots\otimes(F_2\odot P_2)$ where $\odot$ is the element wise (Schur) product?
Is $|x(F_{2^k}\odot P_{2^k})y'|^2\leq|xF_{2^k}F_{2^k}'y'||xP_{2^k}P_{2^k}'y'|\leq \frac{1}{{k}}$?
Regarding your first question, I can verify that the mixed product rule for the Kronecker product is
$$ (A*B)\otimes(X*Y) = (A\otimes X)*(B\otimes Y) $$ where * can be the Schur, Frobenius, or Matrix product -- and assuming the matrices are of compatible sizes for each kind product.
Not sure about the inequality in your second question.