Mediators are often, but not always, lawyers. When the mediator is picked by agreement, there is some chance of getting a mediator who is experienced in mediating cases like yours. Sometimes the judge assigns the mediator. The ability to find an arbitrator just right for your lawsuit is then lost. In my experience, both sides are better off selecting their own mediator.
Most mediators charge based on the number of parties.
After an opening session where the lawyers make opening statements, the two sides usually go off to separate rooms. The mediator goes back and forth between the rooms relating settlement offers and counter offers, and exploring the strengths and weaknesses of each side's case. Having a client hear a neutral stranger discuss problems the client is facing is usually very valuable. And, believe me, there is hardly a case where both sides do not have problems. By the end of the day, the case has either settled or each side realizes that full preparation for trial is necessary.