A: There is nothing to clear your name of unless you are criminally charged. It is unclear from your question where the money went. Did it go toward your rent? If so, then perhaps just give her the money back to keep the peace as you would need to pay your rent regardless of whether or not she was living with you.
A: There is no such thing as "full legal and physical custody" in Florida. There needs to be a determination regarding parental responsibility. Then there needs to be a determination regarding a parenting plan and timesharing. These are very specific concepts and permit both parents to actually parent their minor children to the extent possible. In the event you and he do not agree as to these issues, then you would have a trial and the court would make determinations in regard to these issues. I would recommend that you have a consultation with an attorney to discuss in detail how this works and what you can expect moving forward if you and he do not reach an agreement.
A: My colleague is absolutely correct, if the mother filed a petition for protection against domestic violence, the court reviews that and makes an immediate determination without a hearing. In the case that you describe, the court did not grant the injunction and did not set a hearing to extend the injunction. In the event an injunction is granted by the court on what is called an "ex parte" basis, then the court is required to set a hearing within an extremely short time period in order to have a full evidentiary hearing. That did not happen in your case. So, yes, the matter is over. She could appeal the denial of the injunction; however, this is not typically done.