I think so, yes. $q(z)-p(z)$ is a polynomial with epsilon coefficients, we can take epsilon small enough so that on the contour of the region $|q(z)-p(z)|<|p(z)|$. Then appeal to Rouché's theorem to learn that $p(z)$ and $q(z)$ have the same number of zeros in the region.

I think so, yes. Suppose all the roots lie inside the unit disc. By compactness $|p(z)|$ takes a positive minimum $\delta$ on the contour (the unit circle). For $q(z)$ in the epsilon-neighbourhood, $q(z)-p(z)$ is a polynomial with epsilon coefficients, so we can take epsilon small enough so that on the contour $|q(z)-p(z)|<\delta<|p(z)|$. Then appeal to Rouché's theorem to learn that $p(z)$ and $q(z)$ have the same number of zeros in the region.