Florida no longer uses the term "custody." We now have time sharing and parental responsibility. Of course it is possible that you could be granted what you think of as full custody. A lot more information is needed to give you a full evaluation of your chances.
The filing fee for a divorce case varies from county to county, but is about $408.
This is a complicated question and more information is needed to answer it fully. However, we do not use the term "custody" any more we craft parenting plans with time sharing arrangements. You may be able to get sole parental responsibility, but it is not easy. I suggest you contact a local attorney for a consultation. Best of luck, Jennifer Dane
The filing fee is generally $409.00 plus an additional $10.00 if you wish to issue a summons to serve the papers on another individual.
As for the second part to your question, custody and time-sharing are very fact specific issues and unfortunately without additional information, your question is impossible to answer.