They should if the accused is running for public office. And why does it matter if they are unproven? Does that justify the behaviour of Trump and the audience?
That's ridiculous. So if you run for any position of power, anyone can come out with a claim, an unsubstantiated claim and then you're disqualified automatically? That's a dangerous and undemocratic precedent and bound to be abused.
Should Trump have done what he did? Depends on how you look at the situation, did he mock her or the fact that her accusation is taken as fact by anyone on the left? Did he mock a sexual assault victim or a person who expects to be believed even without an ounce of evidence? I agree that he shouldn't have mocked her but she has provided nothing as proof other than "I remember that it was him".