Being fired for harassment/abuse doesn't magically mean that you'll never harass or abuse someone in the future. Hiring someone with a track record of these issues puts your other employees at risk.
Not at all (and I'm not really sure why you're leaping so precipitately to that).
But someone's previous conduct in a professional role is absolutely relevant information when considering hiring them; that's why we have the concept of a reference at all. It's not stigmatising someone or being unfair to them to consider their prior actions.
If someone has previously been fired for harassment, then that doesn't mean that they should never be employed again. Rehabilitation is totally possible, but it's not an automatic thing: someone with that history should expect it to be a concern for future employers, and should be able to outline (as you would with any other issue) the steps they've taken to mitigate it and ensure the same behaviour won't be repeated.
