1. I don't think it had much to do with nationalist sentiment. EU historically had a bad press in the UK, and the greek debacle didn't help. At the same time immigration has been an increasing concern (not just in the UK, everywhere in Europe, and in the US too). The UK has seen a massive inflow (>1 million in 4 years) of eastern european nationals immigrating, which I think is the primary reason why leave won (rightly or wrongly). But none of this is UK specific. The EU is bracing itself for a rise of populist, eu-skeptic parties in the coming elections.
2. I am not sure how you can claim the majority of the populace is against leaving the EU. A national vote demonstrated the opposite. And it's not clear that another vote now wouldn't yield the same result (the recent polls are roughly where they were a few months before the first vote).
3. May is the prime minister. The head of state (the queen) being essentially an honorific title, she is in charge (she has control of the red button!). But parliament has more power in the UK than in the US and she answers to parliament.
4. The terms of trading between the EU and the UK, as agreed with the EU, requires the UK to abide to EU laws without having any say on them. Brexiters are against as they don't accept this loss of sovereignty (they typically would like a free trade deal with the EU but without loss of sovereignty). Remainers are against as they don't really want to leave the EU in the first place, and are trying to force a way out of Brexit.
I don't think you can call the result of referendum "stupid" and ignore it. That would seem very anti-democratic to me. I don't know how the whole thing will play out. Right now it seems that the potential outcomes are:
a. Brexit being postponed from March to June (unlikely to be later as this is when the new EU parliament starts after the spring elections, and unlikely to help in any way other than kicking the can down the road).
b. May being defeated at the vote on Wednesday, new general elections, whoever wins will have to deal with the same problem and likely with the same outcome (the EU is unlikely to concede much).
c. A new referendum (but May repeatedly ruled it out, and again nothing says it will yield another result)
d. The EU making significant enough concessions in the short term that could help PM agreeing to the deal. Germany hinted they would be open, while France ruled it out today.
e. Leaving the EU without a deal, resulting in short term trade disruptions and an adverse economic impact on the medium term.
f. Parliament cancelling Brexit. Many MPs would probably prefer that option, but this is ignoring the result of a referendum, which is a really worrying outcome for a democracy and would likely boost a populist vote down the line.
There aren't any obvious good solution, everything comes at a cost. Which is why the public opinion is terribly divided.