1) Can she ask the court to order the parties to mediation if Jack refuses to go? If so, under what circumstances will the court do so?
Yes, Jill can because the only way that Jack wouldn’t have to attend is if he is authorized by the court or the mediator to attend the mediation by telephone. But other than that he would have to attend the mediation he wouldn’t be able to just ignore it. Stated in M. D. Fla. L.B.R. 9019-2 it says that “(k) Participation of Parties at Mediation . --All parties to the mediation are required to attend the mediation in person, unless authorized by the Court or the mediator to attend by telephone. Parties are encouraged to participate in the mediation in a good faith attempt to resolve the issues between them. Parties who are not individuals shall participate in mediations through the presence of a representative with full authority to settle the matter that is the subject of the mediation.”

2) Who can serve as the mediator (what are the qualifications) if the court directs the mediation?

The qualifications for a mediator in the state of Florida found in M.D. Fla. Local Rule 9.02 is for them”(c) Qualifications of Mediators --An individual may be certified to serve as a mediator if:(1) He or she is a former state court judge who presided in a court of general jurisdiction and was also a member of the bar in the state in which he presided; or(2) He or she is a retired federal judicial officer; or(3) He or she has been a member of a state bar or the bar of the District of Columbia for at least ten (10) years and is currently admitted to the Bar of this Court.In addition, an applicant for certification must have completed a minimum of 40 hours in the Florida Circuit Court Mediation Training Course certified by the Florida Supreme Court and be found competent by the chief judge to perform mediation duties.At the direction of the chief judge, an advisory committee may be constituted to assist in formulating policy and additional standards relating to the qualification of mediators and to assist in reviewing applications of prospective mediators.”

3) If Jack doesn't show up to the mediation proceeding, what can the court do to "convince" Jack to grace the proceeding with his presence?

If Jack does not go to mediation proceeding, he can possibly be fined and will have to make payments for session fees and/or attorney fees. That’s why they try to convince you to go to court so you avoid the fees In Fla. 5th Jud. Cir. AO A2008-27 it states that “Upon the Court's own motion or upon motion of the appearing party or Court Mediation Services, a party's failure to appear for the mediation session may result in sanctions being imposed by the Court against the non-appearing party, including, but not limited to, payment of session fees and attorney fees, if any. These fees will not apply to any future mediation sessions. If an indigent party fails to appear, sanctions, including the cost of a mediator for a session may be applied.” Also Jill can try to convince Jack to attend a class with the mediator to just educate them about the mediation process.

4) If the mediation fails, will the mediator be able to tell the judge what went wrong and how the case might be settled?

No it is supposed to be confidential. Usually the judge won’t even know the mediators name unless it is a court-ordered mediation. Mediation is supposed to remain between the parties, mediator, their lawyer if they have one, and the guest if parties have someone come with them. Whoever is in the mediation whatever is said is supposed to stay between them. Sometimes you have to have a written agreements and inform both parties if a mediator wants to communicate through a written letter to the judge. In M. D. Fla. L.B.R. 9019-2 “(g) Confidentiality .(1) Definitions . --As used in section (g), "Mediation Communication" means an oral or written statement, or nonverbal conduct intended to make an assertion, by or to a participant in a mediation made during the course of the mediation, or prior to a mediation if made in furtherance of a mediation; "Mediation Participant" means a mediation party or a person who attends a mediation in person or by telephone, videoconference, or other electronic means; "Mediation Party" means a person participating in a mediation directly or through a designated representative, and who is a named party, a real party in interest, or who would be a named party or real party in interest if an action relating to the subject matter of the mediation were brought in a court of law; and "Subsequent Proceeding" means an adjudicative process that follows a mediation, including related discovery.M. D. Fla. L.B.R. 9019-2”