Ehh, I'm not sure. The actual right is better described as the right not to be unfairly tried, which is clearly negative. The distinction is more than semantic: if the State launches an investigation and then drops it, leaving suspicion in the public's mind, you can't force them to go to trial to present your case and clear your name.
What is at stake is the liberty of the poor not to be interfered with in taking from the surplus possessions of the rich what is necessary to satisfy their basic needs. Needless to say, libertarians would want to deny that the poor have this liberty. But how could they justify such a denial? As this liberty of the poor has been specified, it is not a positive right to receive something, but a negative right of non-interference.
The right to a fair trial is about two things: you must be tried before being found guilty and punishment/sentencing is administered, and that trial must be executed fairly. In effect, you have the right to a (fair) trial.
A trial and especially a fair trial are undeniably interventions on how things would otherwise proceed. But I agree with moomin that the positive/negative distinction is not really meaningful.
No, because otherwise it would just be aggression. A trial is a method for the accusers to be able to show that their subsequent aggression (imprisonment, removal of property, etc) is justified.
Whether or not you think an "extra right" is necessary has no bearing on why that right is provided.