UK and Ireland are not part of Schengen and neither of them have any intention of joining. Both have separate visa requirements.
Yes; And the Schengen countries can decide which country's citizens may enter Schengen without a visa (for tourism/business/private).
Ukraine, Japan, USA etc. are on a so-called "positive list".
UK is not (according to my recollection). Hence, formally UK-citizens will have the same (or even less) immigration rights into Schengen as Indian, Nigerian or Chinese citizens.