Both were already answered here https://math.stackexchange.com/questions/4389689/is-every-simply-connected-set-in-the-plane-regular-for-brownian-motion

So for Q1 we can use the simple-arc criterion.

>The regularity of every boundary point of an open set in $\mathbb R^2$ is in fact strongly related to connectedness. However, that's more a property of the complement of the set: Problem 4.2.16 in [1] which is:

>*Let $D\subset\mathbb R^2$ be open, and suppose that $a\in\partial D$ has the property that there exists a point $b\not=a$ in $\mathbb R^2\setminus D\,,$ and a simple arc in $\mathbb R^2\setminus D$ connecting $a$ to $b\,.$ Show that $a$ is regular.*

>The solution is provided in [1] section 4.5. The unit disc minus the line segment $[0,1)\times\{0\}$ clearly satisfies the properties of $D$ in that problem.

>[1] I. Karatzas, S. Shreve, *Brownian Motion and Stochastic Calculus.*


More generally for Q2, we can use the article mentioned in the comments
["A remark on the probabilistic solution of the Dirichlet problem
for simply connected domains in the plane"][1]

where they show that: *The Dirichlet problem is solvable for any simply connected domain in $\mathbb{C}$.*



  [1]: https://www.sciencedirect.com/science/article/pii/S0022247X18303433