Yes, we wait for the courts or the legislature rather than resorting to mob justice. There are good reasons for it. The most obvious is to ensure that the corrupt politicians do not use the power of the state to arbitrarily oppress and persecute people. Another is to ensure that there is no doubt about guilt when we convict someone, and to avoid arguments about the penalties being arbitrary and unfair.

Really though, establishing courts to deal with accusations of people breaking the law is something most societies converge on at some point. "Joe was only ordered to spend 5 years working the fields, so why should Jim be executed for doing the same thing????" amounts to a rudimentary form of judicial law -- pointing to a precedent from a previous case to argue the current case.

Yes. This is what we do in the United States and most of Western Europe. This is how countries like this function

Until a judge says it's a crime, it isn't. Innocent until proven guilty etc