This is what I think the relation between people and their government is:
"government of the people, by the people, and for the people"
That was from the Gettysburg address.
For the second part of that question, I'm pretty sure there are more than one type of government in Europe. After all, it is a big continent.
Those are just my opinion.
You need to read what John Locke thought and wrote:
John Locke - Wikipedia, the free encyclopedia John Locke (pronounced /lɒk/; 29 August 1632 – 28 October 1704) was an English philosopher. Locke is considered the first of the British empiricists, ...
John Locke (Stanford Encyclopedia of Philosophy) John Locke (b. 1632, d. 1704) was a British philosopher, Oxford academic and medical researcher, whose association with Anthony Ashley Cooper (later the ...
John Locke John Locke was an Oxford scholar, medical researcher and physician, political operative, economist and idealogue for a revolutionary movement, ...
Social contract - Wikipedia, the free encyclopedia While Rousseau's social contract is based on popular sovereignty and not on individual sovereignty, there are other theories espoused by individualists, ...
Rousseau: Social Contract THE SOCIAL CONTRACT. OR PRINCIPLES OF POLITICAL RIGHT. by Jean Jacques Rousseau. 1762. Translated by G. D. H. Cole, public domain ...
Be sure to look up the term "democracy" -- http://www.answers.com
And yes, "Should it?" is definitely an opinion question. Whether a country should or shouldn't be a democracy, a republic, or whatever is usually determined by the people who live in that country. Whether people in other countries should decide FOR them ... hmmm.