Not all Scots want independence of England, but a great many do.
There are many reasons for it, going all the way back to 1603, when Elizabeth Tudor died and her nearest Protestant relation James VI of Scotland became king of England as well as king of Scotland.
In practically any other country a similar state of affairs would most likely have resulted in the king's original country taking over the new kingdom. James apparently felt uncertain of his welcome in England and chose to move to London. Admittedly the Stuarts were quite often in Scotland after that, but as time went on and the Stuart kings were replaced by the House of Hannover and the Scottish Parliament was disbanded, leaving the former English parliament in London as the UK's only parliament, Scotland began to feel more and more marganilised and ill-done by.
Having grown up in Scotland this brief historical review was part of my schooling, but I early realised that English people know nothing or little of all these facts and genuinely do not understand why Scotland could possibly feel a desire for independence.
An Independant Scotland and the EU?
Well, as Scotland had a majority who voted for remaining in the EU there is good reason to believe that an Independant Scotland will ask to rejoin the EU.
However, the population of Scotland compares quite reasonably to that of Norway and Norway is not a member of the EU, so Scotland might be able to go it alone, although why Scotland would want to, as most Scots didn't want to leave the EU baffles me.
I realise that these political threads tend to lead to mud-slinging, but if you read a little Scottish history south of the border, you might just start to realise how some, perhaps most Scots feel about the whole United Kingdom.
IMO the United Kingdom has never been anything but a ironic statement. Geographically speaking the majority of British citizens who live in the British Isles live in England. This means that the majority of voters live there too, and have never bothered to try and find out how the Scots, Irish (north or south) or Welsh feel and think about the political construction called the United Kingdom.