Kindly ignore my flair for this answer; electoral history is a great hobby of mine!

The concept you're looking for would be called direct democracy; where voters make their minds up on individual issues rather than letting elected officials do so, which is called representative democracy. Most western nations have some kind of mechanism for direct democratic action in the form of referendums or plebiscites, where political representatives empower the public to make a decision on a specific policy issue. In some countries these kind of votes are fairly frequent, in others they are extremely rare. In many countries this power is reserved for particularly big questions - like constitutional amendments or revising the electoral system. There have been some pretty exceptional referenda with profound historic significance, too; in 1992, white South Africans were asked to vote on whether or not the government should continue to negotiate an end to apartheid (and 69% said yes, thankfully!).

The closest thing you will find to a successful, stable direct democracy that has functioned on a large scale would be Switzerland. Switzerland has had numerous policies decided by referendum since the 19th Century; they range from the seemingly obscure and mundane (like a referendum on the match market in 1895, or banning absinthe in 1908) to the profoundly significant (like votes on female suffrage, electoral reform and even economic policy programmes, like in 1935). Historically, local districts have also made decisions by direct democracy, with open-air meetings involving hundreds or thousands of electors debating and verbally voting on local issues. Although today, these open-air sessions have largely given way to private ballots.

Switzerland is however still a representative democracy as well - most legislation passes through an elected legislature without going to a popular vote, and this legislature is elected on the basis of party. Unlike the rest of the western world though, the Swiss do not form governments based on either a single leader or who commands a legislative majority. Switzerland is governed by a Federal Council of multiple people who exercise the power of Head of State together as a democratic body (today it has 7 members); the members of the Federal Council are elected by the legislature and they take turns each year to serve as President, but they are all equal, and government departments are divided up between them. Traditionally, the Federal Council includes representatives from both the left and right wing. So in general, Switzerland is a rather unique democracy.

It is worth noting though that Switzerland perhaps also offers us an example of why this method of governance can be flawed. An attempt was made in 1959 to extend the vote to women at the federal level but rejected by referendum by 67% of (male) voters; it took another 12 years for women to receive equal voting rights at the federal level, when 66% of men voted in favour of an extension of the franchise in 1971. Switzerland was the last large country in Europe to give women the vote on equal terms to men - and at the local level, some more conservative districts still prohibited female suffrage after 1971. Appenzell Innerrhoden did not let women vote in local affairs until 1991, when the Supreme Court ruled enough was enough. Liechtenstein had a similar problem with referendum outcomes obstructing the path to female suffrage until 1984, though Liechtenstein is a tiny nation which makes it an awkward comparator.

Yes. The concept you describe is known as a referendum, where the electoral corps of a country is polled on a specific question, which is either multiple choice or yes or no.

Switzerland is prominent among states for making the most frequent use of these, there having been at least 1 referendum a year respective years of its existence, but others, such as referendums leading up to Australian federation, which were held in each state in the decade leading up to the union in 1901. In 1992, New Zealand changed its voting system following a referendum.

There are plenty of examples out there, frequently on less contentious issues, so rest assured this is neither a new nor novel concept in politics.

The Athenians may have invented democracy, but they thought elections were deeply undemocratic. They figured rich and powerful men would always be able to use their influence and money to procure the desired outcome of an election. This is why they preferred to appoint the holders of government office by lot, and to have important matters decided by taking a vote in the general assembly. Once the Athenian democracy reached its full form in the fourth century BC, it remained remarkably stable for several generations.

Still, there were of course some practical limitations to the system. Firstly, neither women nor foreigners nor slaves could vote, so that Athenian democracy would not count as a democracy by any modern standards. Secondly, all votes on policy were held on a hill called the Pnyx, on which only about 6,000 citizens could fit. Actual decisions were therefore taken by the minority of citizens who could make it in. It helped that by the 4th century BC the Athenian state paid an attendance fee to assembly meetings, so that poverty was no longer a barrier to participation in government, but there is still the fact that significant numbers of Athenian citizens would have lived several days' journey from the city.

It should also be said that, even though there were no formal political parties in Classical Athens, there were definitely factions whose policies tended to follow predictable patterns. These factions tended to form around old aristocratic families or gifted public speakers, who held a recognised disproportionate amount of power in a system based entirely on the ability of politicians to persuade the masses.

This answer (and perhaps the question) is more political science than history; however, one could argue that most constitutional governments follow this model. I'm also hesitant to cite very specific examples die to the 20 year rule.

In the US, particularly at the state level, people can directly vote for ideas that they want. For example, California's propositions are very often concerning a single issue each and generally override the power of the state legislature.

Keep in mind, even if these direct propositions became wildly popular in a state, you would still need some sort of governmental body to administer the propositions and resolve ambiguities that arise (since it works be inefficient to bring every such question to a large body of people). Further, you'd likely want people in those positions that you believe will resolve said issues in the manner you expect so you likely won't be able to escape reviewing the views of elected representatives.