Oh what a can of worms to open!

Unlike Northern Ireland which is neither part of Great Britain or the British Isles. Indeed citizens of Northern Ireland can not strictly refer to them selves as British.
I would agree but I would be careful who you said that too. Depending on their political affiliations it would either earn you a slap on the back or a punch in the face
Lots of wrong bits in it which serve only to muddy the waters. There is no such thing as 'The Republic of Ireland'. My passport and driving licence says Ireland. Sure we are a republic but are we going to start calling the USA: The Republic of the United States of America? It's only used to differentiate us from Northern Ireland which is not a country but at best an artificially created region or territory like the Channel Islands.
One thing that video highlights is how complicated the whole UK thing is. For example Scotland, Northern Ireland and the offshore island have their own banknotes but not Wales. But none of them are legal tender, only Bank of England notes are legal. The Isle of Man and the Channel Islands are British but not part of the UK or members of the EU.
It's a bit of a mess really. Who knows what'll happen is Scotland votes to become independent. Not that I think that will happen. The Scots have got used to sucking off the English teat for too long.