I think it's important to share things and be completely honest with 'the one'. If I was with somebody who had a history, for example, I would want them to share this with me rather than hear it from somebody else. When discussing past relationships however, it is crucial that you do not compare your current partner to previous ones. Unless they are comparatively 'better'. Haha.
This very much depends on your dating profile. If you have a tendency to fall very quickly, and are in love with love, I would say no. Less is more in this instance. Take the time to really get to know who you are involved with, give them the opportunity to get to really know you, and trust will come. If you disclose too soon your romantic interest may feel pressured by too much too soon. Alternatively you may feel embarrassed and vulnerable if the love of your life tends to only last for the month. This of course doesn't include the stuff which has an important place in the here and now, the need to knows that a love interest really need to know up front.
I can't say from personal experience as I'm still in my first serious relationship (which started 22 years ago), but I believe the more you know and understand about each other in a relationship, the better equipped you are as a couple to tackle future challenges together. So "yes", I probably would, and would hope my partner would too, but only from the perspective of understanding each other. Living in the past is a different issue and doesn't really benefit anyone.
I don't think it is necessarily something that needs to be talked about unless it was a long term relationship.
What happened before you met the person, is to some degree not relevant to the current situation, unless of course you are a murderer or some other bad habit.
I think people do, probably as part of talking in terms of what they want out of a relationship (eg marriage, children or not). I wouldn't dwell on the past too much, but would rather talk about what I want in the future.
If your new relationship is serious, then I feel its best to be honest and tell your partner about it. Then again if your past is too irrelevant and nothing much to be discussed about, you may conveniently forget about it