That's the theory, but it's really not true at all in practice.
For one thing, UK MPs are directly elected. If they don't represent their constituents satisfactorily, the constituents can vote them out, personally, next time. In contrast, UK MEPs are elected via a party list system. If they don't represent their constituents satisfactorily, the only way to get rid of the top person on a party's list is for that party not to get enough votes for even a single representative in that region. The personal accountability is lost under this system.
For another thing, UK MPs do tend to communicate meaningfully with their constituents. I have written to my MP on occasion and invariably received at least a sensible reply (and this has been true for several different MPs, not all in the same party). I have visited my MP in person at their surgery and had a face-to-face discussion where I could see they were understanding the issue I was raising. In striking contrast, on the occasions when I have written to my MEPs, I have not received so much as a courtesy acknowledgement from around half, and usually only one or two have replied with any real substance. I have literally never heard (unsolicted) from any MEP representing me other than at election time, I am not aware of any means to meet them face-to-face to discuss an issue, and most of the time I don't even know their names or which parties they represent.
Whatever the pros and cons of the EU in general, the democratic deficit is a legitimate and well-founded criticism.
Personally I think it's more democratic for the government to reflect the will of the people than a local representative to represent the largest minority in an area.
Clearly FPTP is a failure as far as fair representation is concerned. That's not even a subject for intelligent debate, just a mathematical fact. However, it's a different instance of democratic deficit, not the one that people were voting on in this particular case.
EU Parliament doesn't make policy. Therefore the parties have no manifestos worth a damn. You can't "vote for the other guys" if you care about copyright because none of them have any real influence over the Commission which is where all the power truly lies.
If copyright is your main concern, you should look at the Green Party. Julia Reda, linked in this article, is a tireless advocate against copyright overreach.
No, it's not. It's not even a feature of a number of Western representative democracies. (The UK has notably begun in recent years making some steps in the direction of a separation of powers system, but has long been described as having a fusion of powers system.)
In theory, separation of powers puts a brake (but not an absolute barrier) on attempts to convert what is nominally a representative democracy to an autocracy by way of a self-coup by the unitary elected leadership, but it does so directly by making democratic accountability more complex for electors.