> is it true that $\mathbb{P}\{\exists t\in[0,1]:W(t) \in U\}=1$ if and only if $0$ is a regular point for $U$?

Yes, some books even take the definition of regular points to be that eg. see online notes ["Classical potential theory and Brownian motion"][1]. In terms of conditions besides the [Port-Stone book][2], I would look at the Garnett-Marshall book "Harmonic measure" eg. the section on the Wiener-series for regular points in terms of capacities.


  [1]: https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s6_v3_article-09.pdf
  [2]: https://www.sciencedirect.com/book/9780125618502/brownian-motion-and-classical-potential-theory