I agree, these questions are interesting and the answers could be enlightening, so thank you.
As someone who voted leave in 2016 and would be very likely to vote to keep Britain outside the EU if the question were put again in future, my answers are:
...believes the country they live in should remain part of the EU, and if so why, or if not why not? - I live in the UK which is now outside the EU and I am happy that it is. I don't want to see the UK return to EU membership because membership involves cost and I'd prefer my tax £s spent in the UK not elsewhere. I also very much disliked that I couldn't effectively vote to remove the EU's leaders, nor could Britain make her own trade agreements or adequately control her own borders while an EU member state.
whether the individual believes the EU should exist at all?
As someone who lives in a country which is no longer a member I'm cautious to opine on this. However, from my first principles national self-determination is very important to me, If a nation's people wish to collaborate with neighbouring nations under the flag of the EU they should absolutely be able to, Equally, if a nations people decide either to stop such collaboration and leave, or to decline to join (perhaps as a new nation) then those sentiments should be given effect as well.
I found the EU much, much harder to stomach than it's organisational forebear, the EC, because of the increased range of competencies and it being larger, less coherent IMO and consequently with a reduced role for Britain.
Basiically my view is, if the people's of the nations that make up the EU wish it to exist and wish their country to be a member state within it, then it should exist and they should be members.