Process on a state prisoner is the same as you would serve process on any other person. I would advise that you contact the sheriff's department in whatever county the prison is located and arrange effectuating service through them. The Florida court has the power to dissolve a marriage even if the respondent is not a resident of the state of Florida and the court has no personal jurisdiction over the respondent so if that is your only issue the process should be relatively easy.
An attorney that specializes in elder law might be the most appropriate. If not, seek the counsel of a civil litigation lawyer to recover the money via a conversion claim. You may want to make sure the person is collectible before filing suit.
In accordance with Florida Statute, evidence that a parent has been convicted of a misdemeanor of the first degree or higher involving domestic violence creates a rebuttable presumption of detriment to the child; although it is unclear from your question whether any charges were filed resulting in a conviction. Regardless, the court will base it's determination on the best interests of the child. I would recommend seeking help from a parenting coordinator or the appointment of a Guardian Ad Litem.
You can file a motion for continuance and set it for motion calendar before you leave. Alternatively, you can ask your ex if he would be willing to allow you to appear telephonically and submit an agreed order as to same. Unless your father is an attorney, he cannot represent you at the hearing but you can hire a coverage attorney for the hearing in the capacity of a limited appearance for same.
He will have to be served regardless. If you don't know where he went you will have to perform a diligent search, and certify same with the court, before you attempt constructive service. You should get the form for the diligent search from the clerk so you can begin looking for him. However, since he has been gone such a short amount of time I would suggest you call his friends and family and give him a few days to resurface. You have 120 days to serve him after filing.
Benefits stemming from a parents disability are considered income and both parents have a child support obligation, regardless of the circumstances. If he is not reporting income you can attempt to have it imputed to him if you can present evidence that he is accepting money under the table.
Both parents have an obligation to support the child. There will be a parenting plan, including a time sharing schedule, and child support guidelines entered with the final judgment. He may also seek a temporary version of the aforementioned during the pendency of litigation.
If your incomes are relatively equal, or you don't have the ability to finance her legal fees, then you shouldn't be found responsible for her fees unless there is evidence of misconduct which caused the accumulation of unnecessary fees.
Of course not, in fact, you should be commended for this decision! The transition is daunting and seeking help is not only smart, but a step in the right direction to making yourself whole again. FL has a therapist-patient privilege which makes all communications between you confidential absent extraordinary circumstances (ie an event which presents imminent danger to either your health or that of your child's).
You can motion the court for temporary fees and costs based on his ability to pay and your need for same. I am sure there are several excellent attorneys on Avvo that would be willing to help you execute this so that you are able to retain quality legal representation if you do a search specific to Pinellas county.