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.
Britain is the big island, not a country itself.
So to recap:
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The UK and Ireland are both countries.
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The UK is (primarily) made up of England, Scotland, Wales, and Northern Ireland.
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Britain and Ireland are both islands. The Island of Ireland consists of Ireland (the country) and Northern Ireland (part of the UK).
Interesting, I didn't realize I had that wrong! I thought they were all kind of mini countries, not quite meeting the "standard" bar - thanks for the clarifications!