Mediation is your chance to work out an agreement with your opponent outside the courtroom. The two (or more) of you sit down with a trained mediator and hash out what went wrong and what can make it right. Mediation can be used in a variety of situations. For example, you may have lent a chainsaw to a guy who won't give it back. Or you and your spouse are separated and don't want to fight in court over who gets the bedroom set and the hdtv. A mediator is not a judge. She won't decide right or wrong, but will try to help you agree on how to go forward.
Yes, you can, but you're paying a lawyer to think about things that may not occur to you. For example, if you write your own agreement, are you going to be surprised 20 years from now when you retire and your ex shows up to claim a share of your pension?
No. People can't negotiate fairly unless they are on a level playing field. Violence or threatened violence in the relationship makes mediation a bad idea.