I can’t answer most of your questions, but one is easy:

The UK is a country. None of Northern Ireland, England, or Scotland are countries - they’re all a part of the UK.

You may or may not notice an outlier here: Ireland, which is not part of the UK and is its own country. Note that Northern Ireland is part of the UK, and is not a part of Ireland (despite being on the island of Ireland). The UK and the Irish have a… strained relationship.