No, Windsor 3.4 uses Castle Core 3.3, see the release notes quoted in the linked blog post (last bullet point, one of the biggest reasons for the release before Castle Core 4.0 final).
Are these "more critical Windsor bugs highly cited over the last number of years" logged in the Windsor issue tracker? I haven't worked on a project using Windsor for some time so haven't really kept up.
There is some work underway to implement support for .NET Core:
I'm happy to facilitate work people want to do with Windsor, however don't have the time to directly work on it just as others that have worked on it previously no longer have the time or desire. It would be great to have some fresh blood to work through the backlog and progress through to become committers.