Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$, and $(B_t)_{t \ge 0}$ is a Brownian motion.
Yes to both. 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, see Bass "probabilistic techniques in analysis" Prop. II.1.14 (or the article mentioned in the comments "A remark on the probabilistic solution of the Dirichlet problem for simply connected domains in the plane")
where they show that: The Dirichlet problem is solvable for any simply connected domain in $\mathbb{C}$.
