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". In terms of conditions besides the Port book, I would look at the Garnett-Marshall book "Harmonic measure" eg. the section on the Wiener-series for regular points in terms of capacities.

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". In terms of conditions besides the Port-Stone book, I would look at the Garnett-Marshall book "Harmonic measure" eg. the section on the Wiener-series for regular points in terms of capacities.