I'll try to address your posts by number.
1) I do not believe accusations should be discounted out of hand. I think accusations should be reported to the correct authorities and due process should be followed. It's unimportant what I personally feel about someone's testimony unless I'm a juror.
If we want to talk about the troubles of dealing with authorities, I'll probably agree with you on that. That's a different discussion, though.
2) You're using people who one already has a relationship with as an example, and it's muddying your message. It puts a natural bias to believe someone due to your past relationship. For the record, no, I would not believe a family member just because they claimed something. But that's because some members of my family have a habit of stretching the truth or outright fabrication.
To make this question fair, the person telling you the story should be a stranger. In that case, I lack enough information to conclude, but will default to "Have they done what they should to get the ball rolling on a case? If not, give'em some suggestions."
3) I'm with you on that; I think it's possible to be skeptical and take appropriate action once guilt is established, not a moment before. Anything further is vigilante "justice" that has no place in society.
4) see the 2nd paragraph of number 1.