By definition, the referendum gives the power to all the voters and 52% of them said "Leave". So it's just self-evidently incorrect to say that the referendum was "won by a small number of men in politics". It was won by the 17.4 million voters who preferred "Leave", outnumbering the 16.1 million voters who picked "Remain".
The voters have lots of reasons to decide in one way or another. Aside from the voters' values and interests, many arguments matter, sometimes completely accurate arguments, sometimes less accurate ones. The British voters were bombarded by an incredible amount of pro-EU propaganda by the EU and even the British government itself but most of them were able to see through these tricks. The media simply aren't omnipresent.
Concerning the numbered items:
A system is only called democratic when lame excuses and malicious insults of your kind are interpreted as an inconsequential whining of a sore loser – which is exactly how I interpret them. You are clearly singling out a side of the referendum that you disagree with while you would never dare to criticize the Europhiles for their lies, demagogy, fearmongering, threats, and downright blackmailing. A country where a referendum or elections could be questioned or annulled by the government's talking points similar to yours would be a dictatorship and indeed, the history has seen quite a few of examples.
Democracy and freedom are somewhat independent things. A free society guarantees the freedom of press – both pro-EU and anti-EU press (and similarly for every other question). But in a democracy, the press doesn't have any direct control over the political decisions. Again, a system in which someone stands "above the system" and has the power to invalidate people's decisions by some universal insults cannot be called a democracy, by any stretch of imagination.
If people, e.g. campaigners or anyone else, make an incorrect or misleading statement, the people on the other side or in the other campaign may correct them and it's up to the voters to decide who is right. If people make a mistake or decide according to a criterion that is found untrue or at least inaccurate after they vote, the only "fix" compatible with democracy is that the people may learn from their mistakes and avoid repeating them or similar mistakes in the future.
The only exceptions are laws that regulate what kind of misconduct may invalidate elections and referendums. A campaigner's statement that is considered inaccurate a week after the referendum but couldn't have been disproven before the referendum cannot be counted as a factor that invalidates a referendum. And by the way, such inaccurate statements have almost certainly not played an important role, anyway.
To summarize, the "modifications" of the democratic system you propose are not really modifications. They would destroy the very essence of democracy.