Recall that

$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$.

The cardinal $\mathfrak{q}_0$ defined as the smallest cardinality of a subset of $\mathbf{R}$ which is not a $Q$-space.

Q1. Is it true that $MA(\omega_1)$ iff $\omega_1<\mathfrak{p}$ ?

Q2. Is there a model of set theory in which $\omega_1=\mathfrak{q}_0$ and a non-$MA(\omega_1)$ ?